Theorem endisj 6718

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 4055	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 6715	. . 3 |- ( A X. { (/) } ) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 6320	. . . . 5 |- 1o e. On
6	5	elexi 2698	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 6715	. . 3 |- ( B X. { 1o } ) ~~ B
8	3, 7	pm3.2i 270	. 2 |- ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B )
9		xp01disj 6330	. 2 |- ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/)
10		p0ex 4112	. . . 4 |- { (/) } e. _V
11	1, 10	xpex 4654	. . 3 |- ( A X. { (/) } ) e. _V
12	6	snex 4109	. . . 4 |- { 1o } e. _V
13	4, 12	xpex 4654	. . 3 |- ( B X. { 1o } ) e. _V
14		breq1 3932	. . . . 5 |- ( x = ( A X. { (/) } ) -> ( x ~~ A <-> ( A X. { (/) } ) ~~ A ) )
15		breq1 3932	. . . . 5 |- ( y = ( B X. { 1o } ) -> ( y ~~ B <-> ( B X. { 1o } ) ~~ B ) )
16	14, 15	bi2anan9 595	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x ~~ A /\ y ~~ B ) <-> ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) ) )
17		ineq12 3272	. . . . 5 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( x i^i y ) = ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) )
18	17	eqeq1d 2148	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x i^i y ) = (/) <-> ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) )
19	16, 18	anbi12d 464	. . 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 2781	. 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 422	1 |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) )

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686   i^i cin 3070   (/)c0 3363   {csn 3527   class class class wbr 3929   Oncon0 4285   X. cxp 4537   1oc1o 6306   ~~ cen 6632

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-1o 6313  df-en 6635

This theorem is referenced by: (None)