Theorem xpsnen 6787

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 4164	. . 3 |- { B } e. _V
4	1, 3	xpex 4719	. 2 |- ( A X. { B } ) e. _V
5		elxp 4621	. . 3 |- ( y e. ( A X. { B } ) <-> E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) )
6		inteq 3827	. . . . . . . 8 |- ( y = <. x , z >. -> |^| y = |^| <. x , z >. )
7	6	inteqd 3829	. . . . . . 7 |- ( y = <. x , z >. -> |^| |^| y = |^| |^| <. x , z >. )
8		vex 2729	. . . . . . . 8 |- x e. _V
9		vex 2729	. . . . . . . 8 |- z e. _V
10	8, 9	op1stb 4456	. . . . . . 7 |- |^| |^| <. x , z >. = x
11	7, 10	eqtrdi 2215	. . . . . 6 |- ( y = <. x , z >. -> |^| |^| y = x )
12	11, 8	eqeltrdi 2257	. . . . 5 |- ( y = <. x , z >. -> |^| |^| y e. _V )
13	12	adantr 274	. . . 4 |- ( ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) -> |^| |^| y e. _V )
14	13	exlimivv 1884	. . 3 |- ( E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) -> |^| |^| y e. _V )
15	5, 14	sylbi 120	. 2 |- ( y e. ( A X. { B } ) -> |^| |^| y e. _V )
16	8, 2	opex 4207	. . 3 |- <. x , B >. e. _V
17	16	a1i 9	. 2 |- ( x e. A -> <. x , B >. e. _V )
18		eqvisset 2736	. . . . 5 |- ( x = |^| |^| y -> |^| |^| y e. _V )
19		ancom 264	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ x e. A ) /\ z e. { B } ) <-> ( z e. { B } /\ ( y = <. x , z >. /\ x e. A ) ) )
20		anass 399	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ x e. A ) /\ z e. { B } ) <-> ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) )
21		velsn 3593	. . . . . . . . . . . 12 |- ( z e. { B } <-> z = B )
22	21	anbi1i 454	. . . . . . . . . . 11 |- ( ( z e. { B } /\ ( y = <. x , z >. /\ x e. A ) ) <-> ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
23	19, 20, 22	3bitr3i 209	. . . . . . . . . 10 |- ( ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
24	23	exbii 1593	. . . . . . . . 9 |- ( E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> E. z ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
25		opeq2 3759	. . . . . . . . . . . 12 |- ( z = B -> <. x , z >. = <. x , B >. )
26	25	eqeq2d 2177	. . . . . . . . . . 11 |- ( z = B -> ( y = <. x , z >. <-> y = <. x , B >. ) )
27	26	anbi1d 461	. . . . . . . . . 10 |- ( z = B -> ( ( y = <. x , z >. /\ x e. A ) <-> ( y = <. x , B >. /\ x e. A ) ) )
28	2, 27	ceqsexv 2765	. . . . . . . . 9 |- ( E. z ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) <-> ( y = <. x , B >. /\ x e. A ) )
29		inteq 3827	. . . . . . . . . . . . . 14 |- ( y = <. x , B >. -> |^| y = |^| <. x , B >. )
30	29	inteqd 3829	. . . . . . . . . . . . 13 |- ( y = <. x , B >. -> |^| |^| y = |^| |^| <. x , B >. )
31	8, 2	op1stb 4456	. . . . . . . . . . . . 13 |- |^| |^| <. x , B >. = x
32	30, 31	eqtr2di 2216	. . . . . . . . . . . 12 |- ( y = <. x , B >. -> x = |^| |^| y )
33	32	pm4.71ri 390	. . . . . . . . . . 11 |- ( y = <. x , B >. <-> ( x = |^| |^| y /\ y = <. x , B >. ) )
34	33	anbi1i 454	. . . . . . . . . 10 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( ( x = |^| |^| y /\ y = <. x , B >. ) /\ x e. A ) )
35		anass 399	. . . . . . . . . 10 |- ( ( ( x = |^| |^| y /\ y = <. x , B >. ) /\ x e. A ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
36	34, 35	bitri 183	. . . . . . . . 9 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
37	24, 28, 36	3bitri 205	. . . . . . . 8 |- ( E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
38	37	exbii 1593	. . . . . . 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 183	. . . . . 6 |- ( y e. ( A X. { B } ) <-> E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
40		opeq1 3758	. . . . . . . . 9 |- ( x = |^| |^| y -> <. x , B >. = <. |^| |^| y , B >. )
41	40	eqeq2d 2177	. . . . . . . 8 |- ( x = |^| |^| y -> ( y = <. x , B >. <-> y = <. |^| |^| y , B >. ) )
42		eleq1 2229	. . . . . . . 8 |- ( x = |^| |^| y -> ( x e. A <-> |^| |^| y e. A ) )
43	41, 42	anbi12d 465	. . . . . . 7 |- ( x = |^| |^| y -> ( ( y = <. x , B >. /\ x e. A ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
44	43	ceqsexgv 2855	. . . . . 6 |- ( |^| |^| y e. _V -> ( E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
45	39, 44	syl5bb 191	. . . . 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 451	. . 3 |- ( ( y e. ( A X. { B } ) /\ x = |^| |^| y ) <-> ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) )
48	32	adantr 274	. . . . 5 |- ( ( y = <. x , B >. /\ x e. A ) -> x = |^| |^| y )
49	48	pm4.71i 389	. . . 4 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( ( y = <. x , B >. /\ x e. A ) /\ x = |^| |^| y ) )
50	43	pm5.32ri 451	. . . 4 |- ( ( ( y = <. x , B >. /\ x e. A ) /\ x = |^| |^| y ) <-> ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) )
51	49, 50	bitr2i 184	. . 3 |- ( ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) <-> ( y = <. x , B >. /\ x e. A ) )
52		ancom 264	. . 3 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( x e. A /\ y = <. x , B >. ) )
53	47, 51, 52	3bitri 205	. 2 |- ( ( y e. ( A X. { B } ) /\ x = |^| |^| y ) <-> ( x e. A /\ y = <. x , B >. ) )
54	4, 1, 15, 17, 53	en2i 6736	1 |- ( A X. { B } ) ~~ A

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   {csn 3576   <.cop 3579   |^|cint 3824   class class class wbr 3982   X. cxp 4602   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-en 6707

This theorem is referenced by:  xpsneng  6788  endisj  6790