Theorem xpsneng 5495

Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254.

Assertion

Ref	Expression
xpsneng	|- ((A e. C /\ B e. D) -> (A X. {B}) ~~ A)

Proof of Theorem xpsneng

Step	Hyp	Ref	Expression
1		xpeq1 4016	. . 3 |- (x = A -> (x X. {y}) = (A X. {y}))
2		id 73	. . 3 |- (x = A -> x = A)
3	1, 2	breq12d 3351	. 2 |- (x = A -> ((x X. {y}) ~~ x <-> (A X. {y}) ~~ A))
4		sneq 3054	. . . 4 |- (y = B -> {y} = {B})
5		xpeq2 4017	. . . 4 |- ({y} = {B} -> (A X. {y}) = (A X. {B}))
6	4, 5	syl 12	. . 3 |- (y = B -> (A X. {y}) = (A X. {B}))
7	6	breq1d 3348	. 2 |- (y = B -> ((A X. {y}) ~~ A <-> (A X. {B}) ~~ A))
8		visset 2295	. . 3 |- x e. _V
9		visset 2295	. . 3 |- y e. _V
10	8, 9	xpsnen 5494	. 2 |- (x X. {y}) ~~ x
11	3, 7, 10	vtocl2g 2349	1 |- ((A e. C /\ B e. D) -> (A X. {B}) ~~ A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {csn 3044   class class class wbr 3338   X. cxp 3984   ~~ cen 5423

This theorem is referenced by:  cdafi 6086  tarsuc2 15245

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-en 5427

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