Theorem zfpair 2777

Description: The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms.

This theorem should not be referenced by any proof other than axpr 2778. Instead, use zfpair2 2780 below so that the uses of the Axiom of Pairing can be more easily identified.

Assertion

Ref	Expression
zfpair	|- {x, y} e. V

Proof of Theorem zfpair

Step	Hyp	Ref	Expression
1		dfpr2 2422	. 2 |- {x, y} = {w | (w = x \/ w = y)}
2		19.43 1088	. . . . 5 |- (E.z((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> (E.z(z = (/) /\ w = x) \/ E.z(z = {(/)} /\ w = y)))
3		prlem2 771	. . . . . 6 |- (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> ((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y))))
4	3	exbii 1051	. . . . 5 |- (E.z((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y))))
5		19.41v 1305	. . . . . . 7 |- (E.z(z = (/) /\ w = x) <-> (E.z z = (/) /\ w = x))
6		0ex 2711	. . . . . . . 8 |- (/) e. V
7	6	isseti 1815	. . . . . . 7 |- E.z z = (/)
8	5, 7	mpbiran 728	. . . . . 6 |- (E.z(z = (/) /\ w = x) <-> w = x)
9		19.41v 1305	. . . . . . 7 |- (E.z(z = {(/)} /\ w = y) <-> (E.z z = {(/)} /\ w = y))
10		p0ex 2770	. . . . . . . 8 |- {(/)} e. V
11	10	isseti 1815	. . . . . . 7 |- E.z z = {(/)}
12	9, 11	mpbiran 728	. . . . . 6 |- (E.z(z = {(/)} /\ w = y) <-> w = y)
13	8, 12	orbi12i 257	. . . . 5 |- ((E.z(z = (/) /\ w = x) \/ E.z(z = {(/)} /\ w = y)) <-> (w = x \/ w = y))
14	2, 4, 13	3bitr3r 182	. . . 4 |- ((w = x \/ w = y) <-> E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y))))
15	14	abbii 1575	. . 3 |- {w | (w = x \/ w = y)} = {w | E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)))}
16		dfpr2 2422	. . . . 5 |- {(/), {(/)}} = {z | (z = (/) \/ z = {(/)})}
17		pp0ex 2771	. . . . 5 |- {(/), {(/)}} e. V
18	16, 17	eqeltrr 1545	. . . 4 |- {z | (z = (/) \/ z = {(/)})} e. V
19		equequ2 1135	. . . . . . . 8 |- (v = x -> (w = v <-> w = x))
20		0inp0 2738	. . . . . . . 8 |- (z = (/) -> -. z = {(/)})
21	19, 20	prlem1 770	. . . . . . 7 |- (v = x -> (z = (/) -> (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
22	21	19.21adv 1288	. . . . . 6 |- (v = x -> (z = (/) -> A.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
23	22	a4imev 1273	. . . . 5 |- (z = (/) -> E.vA.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v))
24		equequ2 1135	. . . . . . . . 9 |- (v = y -> (w = v <-> w = y))
25	20	con2i 97	. . . . . . . . 9 |- (z = {(/)} -> -. z = (/))
26	24, 25	prlem1 770	. . . . . . . 8 |- (v = y -> (z = {(/)} -> (((z = {(/)} /\ w = y) \/ (z = (/) /\ w = x)) -> w = v)))
27		orcom 246	. . . . . . . 8 |- (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) <-> ((z = {(/)} /\ w = y) \/ (z = (/) /\ w = x)))
28	26, 27	syl7ib 216	. . . . . . 7 |- (v = y -> (z = {(/)} -> (((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
29	28	19.21adv 1288	. . . . . 6 |- (v = y -> (z = {(/)} -> A.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v)))
30	29	a4imev 1273	. . . . 5 |- (z = {(/)} -> E.vA.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v))
31	23, 30	jaoi 341	. . . 4 |- ((z = (/) \/ z = {(/)}) -> E.vA.w(((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)) -> w = v))
32	18, 31	zfrep4 2701	. . 3 |- {w | E.z((z = (/) \/ z = {(/)}) /\ ((z = (/) /\ w = x) \/ (z = {(/)} /\ w = y)))} e. V
33	15, 32	eqeltr 1544	. 2 |- {w | (w = x \/ w = y)} e. V
34	1, 33	eqeltr 1544	1 |- {x, y} e. V

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811  (/)c0 2280  {csn 2409  {cpr 2410

This theorem is referenced by:  axpr 2778

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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