Theorem endisj 7122

Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)

Hypotheses

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

Assertion

Ref	Expression
endisj	|- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) )

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem endisj

Step	Hyp	Ref	Expression
1		endisj.1	. . . 4 |- A e. _V
2		0ex 4260	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 7119	. . 3 |- ( A X. { (/) } ) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 6694	. . . . 5 |- 1o e. On
6	5	elexi 2834	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 7119	. . 3 |- ( B X. { 1o } ) ~~ B
8	3, 7	pm3.2i 272	. 2 |- ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B )
9		xp01disj 6706	. 2 |- ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/)
10		p0ex 4325	. . . 4 |- { (/) } e. _V
11	1, 10	xpex 4891	. . 3 |- ( A X. { (/) } ) e. _V
12	6	snex 4322	. . . 4 |- { 1o } e. _V
13	4, 12	xpex 4891	. . 3 |- ( B X. { 1o } ) e. _V
14		breq1 4133	. . . . 5 |- ( x = ( A X. { (/) } ) -> ( x ~~ A <-> ( A X. { (/) } ) ~~ A ) )
15		breq1 4133	. . . . 5 |- ( y = ( B X. { 1o } ) -> ( y ~~ B <-> ( B X. { 1o } ) ~~ B ) )
16	14, 15	bi2anan9 614	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x ~~ A /\ y ~~ B ) <-> ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) ) )
17		ineq12 3427	. . . . 5 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( x i^i y ) = ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) )
18	17	eqeq1d 2247	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x i^i y ) = (/) <-> ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) )
19	16, 18	anbi12d 477	. . 3 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) ) <-> ( ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) /\ ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) ) )
20	11, 13, 19	spc2ev 2921	. 2 |- ( ( ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) /\ ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) -> E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) ) )
21	8, 9, 20	mp2an 430	1 |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   i^i cin 3219   (/)c0 3520   {csn 3709   class class class wbr 4130   Oncon0 4508   X. cxp 4772   1oc1o 6680   ~~ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-1o 6687  df-en 7023

This theorem is used by: (None)