Theorem xpsnen 7085

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 4303	. . 3 |- { B } e. _V
4	1, 3	xpex 4871	. 2 |- ( A X. { B } ) e. _V
5		elxp 4771	. . 3 |- ( y e. ( A X. { B } ) <-> E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) )
6		inteq 3957	. . . . . . . 8 |- ( y = <. x , z >. -> |^| y = |^| <. x , z >. )
7	6	inteqd 3959	. . . . . . 7 |- ( y = <. x , z >. -> |^| |^| y = |^| |^| <. x , z >. )
8		vex 2818	. . . . . . . 8 |- x e. _V
9		vex 2818	. . . . . . . 8 |- z e. _V
10	8, 9	op1stb 4604	. . . . . . 7 |- |^| |^| <. x , z >. = x
11	7, 10	eqtrdi 2283	. . . . . 6 |- ( y = <. x , z >. -> |^| |^| y = x )
12	11, 8	eqeltrdi 2325	. . . . 5 |- ( y = <. x , z >. -> |^| |^| y e. _V )
13	12	adantr 276	. . . 4 |- ( ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) -> |^| |^| y e. _V )
14	13	exlimivv 1948	. . 3 |- ( E. x E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) -> |^| |^| y e. _V )
15	5, 14	sylbi 121	. 2 |- ( y e. ( A X. { B } ) -> |^| |^| y e. _V )
16	8, 2	opex 4350	. . 3 |- <. x , B >. e. _V
17	16	a1i 9	. 2 |- ( x e. A -> <. x , B >. e. _V )
18		eqvisset 2826	. . . . 5 |- ( x = |^| |^| y -> |^| |^| y e. _V )
19		ancom 266	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ x e. A ) /\ z e. { B } ) <-> ( z e. { B } /\ ( y = <. x , z >. /\ x e. A ) ) )
20		anass 401	. . . . . . . . . . 11 |- ( ( ( y = <. x , z >. /\ x e. A ) /\ z e. { B } ) <-> ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) )
21		velsn 3711	. . . . . . . . . . . 12 |- ( z e. { B } <-> z = B )
22	21	anbi1i 458	. . . . . . . . . . 11 |- ( ( z e. { B } /\ ( y = <. x , z >. /\ x e. A ) ) <-> ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
23	19, 20, 22	3bitr3i 210	. . . . . . . . . 10 |- ( ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
24	23	exbii 1654	. . . . . . . . 9 |- ( E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> E. z ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) )
25		opeq2 3889	. . . . . . . . . . . 12 |- ( z = B -> <. x , z >. = <. x , B >. )
26	25	eqeq2d 2246	. . . . . . . . . . 11 |- ( z = B -> ( y = <. x , z >. <-> y = <. x , B >. ) )
27	26	anbi1d 465	. . . . . . . . . 10 |- ( z = B -> ( ( y = <. x , z >. /\ x e. A ) <-> ( y = <. x , B >. /\ x e. A ) ) )
28	2, 27	ceqsexv 2855	. . . . . . . . 9 |- ( E. z ( z = B /\ ( y = <. x , z >. /\ x e. A ) ) <-> ( y = <. x , B >. /\ x e. A ) )
29		inteq 3957	. . . . . . . . . . . . . 14 |- ( y = <. x , B >. -> |^| y = |^| <. x , B >. )
30	29	inteqd 3959	. . . . . . . . . . . . 13 |- ( y = <. x , B >. -> |^| |^| y = |^| |^| <. x , B >. )
31	8, 2	op1stb 4604	. . . . . . . . . . . . 13 |- |^| |^| <. x , B >. = x
32	30, 31	eqtr2di 2284	. . . . . . . . . . . 12 |- ( y = <. x , B >. -> x = |^| |^| y )
33	32	pm4.71ri 392	. . . . . . . . . . 11 |- ( y = <. x , B >. <-> ( x = |^| |^| y /\ y = <. x , B >. ) )
34	33	anbi1i 458	. . . . . . . . . 10 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( ( x = |^| |^| y /\ y = <. x , B >. ) /\ x e. A ) )
35		anass 401	. . . . . . . . . 10 |- ( ( ( x = |^| |^| y /\ y = <. x , B >. ) /\ x e. A ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
36	34, 35	bitri 184	. . . . . . . . 9 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
37	24, 28, 36	3bitri 206	. . . . . . . 8 |- ( E. z ( y = <. x , z >. /\ ( x e. A /\ z e. { B } ) ) <-> ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
38	37	exbii 1654	. . . . . . 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 184	. . . . . 6 |- ( y e. ( A X. { B } ) <-> E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) )
40		opeq1 3888	. . . . . . . . 9 |- ( x = |^| |^| y -> <. x , B >. = <. |^| |^| y , B >. )
41	40	eqeq2d 2246	. . . . . . . 8 |- ( x = |^| |^| y -> ( y = <. x , B >. <-> y = <. |^| |^| y , B >. ) )
42		eleq1 2297	. . . . . . . 8 |- ( x = |^| |^| y -> ( x e. A <-> |^| |^| y e. A ) )
43	41, 42	anbi12d 473	. . . . . . 7 |- ( x = |^| |^| y -> ( ( y = <. x , B >. /\ x e. A ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
44	43	ceqsexgv 2949	. . . . . 6 |- ( |^| |^| y e. _V -> ( E. x ( x = |^| |^| y /\ ( y = <. x , B >. /\ x e. A ) ) <-> ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) ) )
45	39, 44	bitrid 192	. . . . 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 455	. . 3 |- ( ( y e. ( A X. { B } ) /\ x = |^| |^| y ) <-> ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) )
48	32	adantr 276	. . . . 5 |- ( ( y = <. x , B >. /\ x e. A ) -> x = |^| |^| y )
49	48	pm4.71i 391	. . . 4 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( ( y = <. x , B >. /\ x e. A ) /\ x = |^| |^| y ) )
50	43	pm5.32ri 455	. . . 4 |- ( ( ( y = <. x , B >. /\ x e. A ) /\ x = |^| |^| y ) <-> ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) )
51	49, 50	bitr2i 185	. . 3 |- ( ( ( y = <. |^| |^| y , B >. /\ |^| |^| y e. A ) /\ x = |^| |^| y ) <-> ( y = <. x , B >. /\ x e. A ) )
52		ancom 266	. . 3 |- ( ( y = <. x , B >. /\ x e. A ) <-> ( x e. A /\ y = <. x , B >. ) )
53	47, 51, 52	3bitri 206	. 2 |- ( ( y e. ( A X. { B } ) /\ x = |^| |^| y ) <-> ( x e. A /\ y = <. x , B >. ) )
54	4, 1, 15, 17, 53	en2i 7022	1 |- ( A X. { B } ) ~~ A

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   |^|cint 3954   class class class wbr 4114   X. cxp 4752   ~~ cen 6986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-en 6989

This theorem is referenced by:  xpsneng  7086  endisj  7088