Theorem xpsnen 6318

Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

Ref	Expression
xpsnen.1	|- A e. _V
xpsnen.2	|- B e. _V

Assertion

Ref	Expression
xpsnen	|- ( A X. { B } ) ~~ A

Proof of Theorem xpsnen

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsnen.1	. . 3 |- A e. _V
2		xpsnen.2	. . . 4 |- B e. _V
3	2	snex 3957	. . 3 |- { B } e. _V
4	1, 3	xpex 4471	. 2 |- ( A X. { B } ) e. _V
5		elxp 4380	. . 3 |- ( y e. ( A X. { B } ) <-> E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) )
6		inteq 3639	. . . . . . . 8 |- ( y = <. x , z >. -> |^| y = |^| <. x , z >. )
7	6	inteqd 3641	. . . . . . 7 |- ( y = <. x , z >. -> |^| |^| y = |^| |^| <. x , z >. )
8		vex 2604	. . . . . . . 8 |- x e. _V
9		vex 2604	. . . . . . . 8 |- z e. _V
10	8, 9	op1stb 4227	. . . . . . 7 |- |^| |^| <. x , z >. = x
11	7, 10	syl6eq 2129	. . . . . 6 |- ( y = <. x , z >. -> |^| |^| y = x )
12	11, 8	syl6eqel 2169	. . . . 5 |- ( y = <. x , z >. -> |^| |^| y e. _V )
13	12	adantr 270	. . . 4 |- ( ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) -> |^| |^| y e. _V )
14	13	exlimivv 1817	. . 3 |- ( E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) -> |^| |^| y e. _V )
15	5, 14	sylbi 119	. 2 |- ( y e. ( A X. { B } ) -> |^| |^| y e. _V )
16	8, 2	opex 3984	. . 3 |- <. x , B >. e. _V
17	16	a1i 9	. 2 |- ( x e. A -> <. x , B >. e. _V )
18		eqvisset 2609	. . . . 5 |- ( x = |^| |^| y -> |^| |^| y e. _V )
19		ancom 262	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ x e. A ) /\ z e. { B } ) <-> ( z e. { B } /\ ( y = <. x , z >. /\ x e. A ) ) )
20		anass 393	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ x e. A ) /\ z e. { B } ) <-> ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) )
21		velsn 3415	. . . . . . . . . . . 12 |- ( z e. { B } <-> z = B )
22	21	anbi1i 445	. . . . . . . . . . 11 |- ( ( z e. { B } /\ ( y = <. x , z >. /\ x e. A ) ) <-> ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
23	19, 20, 22	3bitr3i 208	. . . . . . . . . 10 |- ( ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
24	23	exbii 1536	. . . . . . . . 9 |- ( E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> E. z ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
25		opeq2 3571	. . . . . . . . . . . 12 |- ( z = B -> <. x , z >. = <. x , B >. )
26	25	eqeq2d 2092	. . . . . . . . . . 11 |- ( z = B -> ( y = <. x , z >. <-> y = <. x , B >. ) )
27	26	anbi1d 452	. . . . . . . . . 10 |- ( z = B -> ( ( y = <. x , z >. /\ x e. A ) <-> ( y = <. x , B >. /\ x e. A ) ) )
28	2, 27	ceqsexv 2638	. . . . . . . . 9 |- ( E. z ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) <-> ( y = <. x , B >. /\ x e. A ) )
29		inteq 3639	. . . . . . . . . . . . . 14 |- ( y = <. x , B >. -> |^| y = |^| <. x , B >. )
30	29	inteqd 3641	. . . . . . . . . . . . 13 |- ( y = <. x , B >. -> |^| |^| y = |^| |^| <. x , B >. )
31	8, 2	op1stb 4227	. . . . . . . . . . . . 13 |- |^| |^| <. x , B >. = x
32	30, 31	syl6req 2130	. . . . . . . . . . . 12 |- ( y = <. x , B >. -> x = |^| |^| y )
33	32	pm4.71ri 384	. . . . . . . . . . 11 |- ( y = <. x , B >. <-> ( x = |^| |^| y /\ y = <. x , B >. ) )
34	33	anbi1i 445	. . . . . . . . . 10 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( ( x = |^| |^| y /\ y = <. x , B >. ) /\ x e. A ) )
35		anass 393	. . . . . . . . . 10 |- ( ( ( x = |^| |^| y /\ y = <. x , B >. ) /\ x e. A ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
36	34, 35	bitri 182	. . . . . . . . 9 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
37	24, 28, 36	3bitri 204	. . . . . . . 8 |- ( E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
38	37	exbii 1536	. . . . . . 7 |- ( E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
39	5, 38	bitri 182	. . . . . 6 |- ( y e. ( A X. { B } ) <-> E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
40		opeq1 3570	. . . . . . . . 9 |- ( x = |^| |^| y -> <. x , B >. = <. |^| |^| y , B >. )
41	40	eqeq2d 2092	. . . . . . . 8 |- ( x = |^| |^| y -> ( y = <. x , B >. <-> y = <. |^| |^| y , B >. ) )
42		eleq1 2141	. . . . . . . 8 |- ( x = |^| |^| y -> ( x e. A <-> |^| |^| y e. A ) )
43	41, 42	anbi12d 456	. . . . . . 7 |- ( x = |^| |^| y -> ( ( y = <. x , B >. /\ x e. A ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
44	43	ceqsexgv 2724	. . . . . 6 |- ( |^| |^| y e. _V -> ( E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
45	39, 44	syl5bb 190	. . . . 5 |- ( |^| |^| y e. _V -> ( y e. ( A X. { B } ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
46	18, 45	syl 14	. . . 4 |- ( x = |^| |^| y -> ( y e. ( A X. { B } ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
47	46	pm5.32ri 442	. . 3 |- ( ( y e. ( A X. { B } ) /\ x = |^| |^| y ) <-> ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) )
48	32	adantr 270	. . . . 5 |- ( ( y = <. x , B >. /\ x e. A ) -> x = |^| |^| y )
49	48	pm4.71i 383	. . . 4 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( ( y = <. x , B >. /\ x e. A ) /\ x = |^| |^| y ) )
50	43	pm5.32ri 442	. . . 4 |- ( ( ( y = <. x , B >. /\ x e. A ) /\ x = |^| |^| y ) <-> ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) )
51	49, 50	bitr2i 183	. . 3 |- ( ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) <-> ( y = <. x , B >. /\ x e. A ) )
52		ancom 262	. . 3 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( x e. A /\ y = <. x , B >. ) )
53	47, 51, 52	3bitri 204	. 2 |- ( ( y e. ( A X. { B } ) /\ x = |^| |^| y ) <-> ( x e. A /\ y = <. x , B >. ) )
54	4, 1, 15, 17, 53	en2i 6273	1 |- ( A X. { B } ) ~~ A

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   {csn 3398   <.cop 3401   |^|cint 3636   class class class wbr 3785   X. cxp 4361   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-en 6245

This theorem is referenced by:  xpsneng  6319  endisj  6321