Theorem tfi 3133

Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A.

See theorem tfindes 3171 or tfinds 3168 for the version involving basis and induction hypotheses.

Assertion

Ref	Expression
tfi	|- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> A = On)

Distinct variable group:   x,A

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 2167	. . . . . . . . 9 |- (x e. (On \ A) -> -. x e. A)
2	1	adantl 390	. . . . . . . 8 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> -. x e. A)
3		difin0ss 2337	. . . . . . . . . . . . 13 |- (((On \ A) i^i x) = (/) -> (x (_ On -> x (_ A))
4		onsst 2999	. . . . . . . . . . . . 13 |- (x e. On -> x (_ On)
5	3, 4	syl5com 52	. . . . . . . . . . . 12 |- (x e. On -> (((On \ A) i^i x) = (/) -> x (_ A))
6	5	imim1d 28	. . . . . . . . . . 11 |- (x e. On -> ((x (_ A -> x e. A) -> (((On \ A) i^i x) = (/) -> x e. A)))
7	6	a2i 9	. . . . . . . . . 10 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. On -> (((On \ A) i^i x) = (/) -> x e. A)))
8		eldifi 2166	. . . . . . . . . 10 |- (x e. (On \ A) -> x e. On)
9	7, 8	syl5 21	. . . . . . . . 9 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. (On \ A) -> (((On \ A) i^i x) = (/) -> x e. A)))
10	9	imp 350	. . . . . . . 8 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> (((On \ A) i^i x) = (/) -> x e. A))
11	2, 10	mtod 108	. . . . . . 7 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> -. ((On \ A) i^i x) = (/))
12	11	ex 373	. . . . . 6 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. (On \ A) -> -. ((On \ A) i^i x) = (/)))
13	12	r19.20i2 1706	. . . . 5 |- (A.x e. On (x (_ A -> x e. A) -> A.x e. (On \ A) -. ((On \ A) i^i x) = (/))
14		ralnex 1656	. . . . 5 |- (A.x e. (On \ A) -. ((On \ A) i^i x) = (/) <-> -. E.x e. (On \ A)((On \ A) i^i x) = (/))
15	13, 14	sylib 198	. . . 4 |- (A.x e. On (x (_ A -> x e. A) -> -. E.x e. (On \ A)((On \ A) i^i x) = (/))
16		ssdif0 2332	. . . . . 6 |- (On (_ A <-> (On \ A) = (/))
17	16	necon3bbii 1600	. . . . 5 |- (-. On (_ A <-> (On \ A) =/= (/))
18		ordon 2994	. . . . . 6 |- Ord On
19		difss 2171	. . . . . 6 |- (On \ A) (_ On
20		tz7.5 2976	. . . . . 6 |- ((Ord On /\ (On \ A) (_ On /\ (On \ A) =/= (/)) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
21	18, 19, 20	mp3an12 908	. . . . 5 |- ((On \ A) =/= (/) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
22	17, 21	sylbi 199	. . . 4 |- (-. On (_ A -> E.x e. (On \ A)((On \ A) i^i x) = (/))
23	15, 22	nsyl2 118	. . 3 |- (A.x e. On (x (_ A -> x e. A) -> On (_ A)
24	23	anim2i 335	. 2 |- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> (A (_ On /\ On (_ A))
25		eqss 2081	. 2 |- (A = On <-> (A (_ On /\ On (_ A))
26	24, 25	sylibr 200	1 |- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> A = On)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  E.wrex 1649   \ cdif 2048   i^i cin 2050   (_ wss 2051  (/)c0 2284  Ord word 2954  Oncon0 2955

This theorem is referenced by:  tfis 3134

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959

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