Theorem xpen 4468

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 4395	. 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 4422	. . . . 5 |- (A ~<_ B -> (C X. A) ~<_ (C X. B))
5		xpen.1	. . . . . 6 |- A e. V
6	5, 3	xpdom2 4422	. . . . 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 4438	. . . . 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 3250	. . . . 5 |- (C X. B) e. V
11		sbthbg 4438	. . . . 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 3250	. . . . 5 |- (B X. C) e. V
15	3, 2	xpcomen 4419	. . . . 5 |- (C X. B) ~~ (B X. C)
16		enen2 4458	. . . . 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 695	. . . 4 |- ((C X. A) ~~ (C X. B) <-> (C X. A) ~~ (B X. C))
18	5, 3	xpex 3250	. . . . 5 |- (A X. C) e. V
19	3, 5	xpcomen 4419	. . . . 5 |- (C X. A) ~~ (A X. C)
20		enen1 4457	. . . . 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 695	. . . 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 4422	. . . 4 |- (C ~<_ D -> (B X. C) ~<_ (B X. D))
26	3, 2	xpdom2 4422	. . . 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 4438	. . . 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 3250	. . . 4 |- (B X. D) e. V
31		sbthbg 4438	. . . 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 955  Vcvv 1802   class class class wbr 2609   X. cxp 3158   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  unxpdom2 4817  sucxpdom 4818  cdaassen 4902  mapcdaen 4904  xpomen 7442  qnnen 7446  infxpidmlem1 7495  infxpidmlem10 7504  infxpidmlem12 7506

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-er 4245  df-en 4351  df-dom 4352

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