Theorem xpen 4488

Description: Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpen.1	|- A e. V
xpen.2	|- B e. V
xpen.3	|- C e. V
xpen.4	|- D e. V

Assertion

Ref	Expression
xpen	|- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))

Proof of Theorem xpen

Step	Hyp	Ref	Expression
1		entrt 4414	. 2 |- (((A X. C) ~~ (B X. C) /\ (B X. C) ~~ (B X. D)) -> (A X. C) ~~ (B X. D))
2		xpen.2	. . . . . 6 |- B e. V
3		xpen.3	. . . . . 6 |- C e. V
4	2, 3	xpdom2 4442	. . . . 5 |- (A ~<_ B -> (C X. A) ~<_ (C X. B))
5		xpen.1	. . . . . 6 |- A e. V
6	5, 3	xpdom2 4442	. . . . 5 |- (B ~<_ A -> (C X. B) ~<_ (C X. A))
7	4, 6	anim12i 333	. . . 4 |- ((A ~<_ B /\ B ~<_ A) -> ((C X. A) ~<_ (C X. B) /\ (C X. B) ~<_ (C X. A)))
8		sbthbg 4458	. . . . 5 |- (B e. V -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
9	2, 8	ax-mp 7	. . . 4 |- ((A ~<_ B /\ B ~<_ A) <-> A ~~ B)
10	3, 2	xpex 3260	. . . . 5 |- (C X. B) e. V
11		sbthbg 4458	. . . . 5 |- ((C X. B) e. V -> (((C X. A) ~<_ (C X. B) /\ (C X. B) ~<_ (C X. A)) <-> (C X. A) ~~ (C X. B)))
12	10, 11	ax-mp 7	. . . 4 |- (((C X. A) ~<_ (C X. B) /\ (C X. B) ~<_ (C X. A)) <-> (C X. A) ~~ (C X. B))
13	7, 9, 12	3imtr3 218	. . 3 |- (A ~~ B -> (C X. A) ~~ (C X. B))
14	2, 3	xpex 3260	. . . . 5 |- (B X. C) e. V
15	3, 2	xpcomen 4439	. . . . 5 |- (C X. B) ~~ (B X. C)
16		enen2 4478	. . . . 5 |- (((B X. C) e. V /\ (C X. B) ~~ (B X. C)) -> ((C X. A) ~~ (C X. B) <-> (C X. A) ~~ (B X. C)))
17	14, 15, 16	mp2an 697	. . . 4 |- ((C X. A) ~~ (C X. B) <-> (C X. A) ~~ (B X. C))
18	5, 3	xpex 3260	. . . . 5 |- (A X. C) e. V
19	3, 5	xpcomen 4439	. . . . 5 |- (C X. A) ~~ (A X. C)
20		enen1 4477	. . . . 5 |- (((A X. C) e. V /\ (C X. A) ~~ (A X. C)) -> ((C X. A) ~~ (B X. C) <-> (A X. C) ~~ (B X. C)))
21	18, 19, 20	mp2an 697	. . . 4 |- ((C X. A) ~~ (B X. C) <-> (A X. C) ~~ (B X. C))
22	17, 21	bitr 173	. . 3 |- ((C X. A) ~~ (C X. B) <-> (A X. C) ~~ (B X. C))
23	13, 22	sylib 198	. 2 |- (A ~~ B -> (A X. C) ~~ (B X. C))
24		xpen.4	. . . . 5 |- D e. V
25	24, 2	xpdom2 4442	. . . 4 |- (C ~<_ D -> (B X. C) ~<_ (B X. D))
26	3, 2	xpdom2 4442	. . . 4 |- (D ~<_ C -> (B X. D) ~<_ (B X. C))
27	25, 26	anim12i 333	. . 3 |- ((C ~<_ D /\ D ~<_ C) -> ((B X. C) ~<_ (B X. D) /\ (B X. D) ~<_ (B X. C)))
28		sbthbg 4458	. . . 4 |- (D e. V -> ((C ~<_ D /\ D ~<_ C) <-> C ~~ D))
29	24, 28	ax-mp 7	. . 3 |- ((C ~<_ D /\ D ~<_ C) <-> C ~~ D)
30	2, 24	xpex 3260	. . . 4 |- (B X. D) e. V
31		sbthbg 4458	. . . 4 |- ((B X. D) e. V -> (((B X. C) ~<_ (B X. D) /\ (B X. D) ~<_ (B X. C)) <-> (B X. C) ~~ (B X. D)))
32	30, 31	ax-mp 7	. . 3 |- (((B X. C) ~<_ (B X. D) /\ (B X. D) ~<_ (B X. C)) <-> (B X. C) ~~ (B X. D))
33	27, 29, 32	3imtr3 218	. 2 |- (C ~~ D -> (B X. C) ~~ (B X. D))
34	1, 23, 33	syl2an 454	1 |- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958  Vcvv 1811   class class class wbr 2619   X. cxp 3168   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  unxpdom2 4845  sucxpdom 4846  cdaassen 4930  mapcdaen 4932  xpomen 7500  qnnen 7503  infxpidmlem1 7552  infxpidmlem10 7561  infxpidmlem12 7563

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-9 965  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  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-ral 1649  df-rex 1650  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  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369

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