Theorem endisj 6802

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 4116	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 6799	. . 3 |- ( A X. { (/) } ) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 6402	. . . . 5 |- 1o e. On
6	5	elexi 2742	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 6799	. . 3 |- ( B X. { 1o } ) ~~ B
8	3, 7	pm3.2i 270	. 2 |- ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B )
9		xp01disj 6412	. 2 |- ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/)
10		p0ex 4174	. . . 4 |- { (/) } e. _V
11	1, 10	xpex 4726	. . 3 |- ( A X. { (/) } ) e. _V
12	6	snex 4171	. . . 4 |- { 1o } e. _V
13	4, 12	xpex 4726	. . 3 |- ( B X. { 1o } ) e. _V
14		breq1 3992	. . . . 5 |- ( x = ( A X. { (/) } ) -> ( x ~~ A <-> ( A X. { (/) } ) ~~ A ) )
15		breq1 3992	. . . . 5 |- ( y = ( B X. { 1o } ) -> ( y ~~ B <-> ( B X. { 1o } ) ~~ B ) )
16	14, 15	bi2anan9 601	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x ~~ A /\ y ~~ B ) <-> ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) ) )
17		ineq12 3323	. . . . 5 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( x i^i y ) = ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) )
18	17	eqeq1d 2179	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x i^i y ) = (/) <-> ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) )
19	16, 18	anbi12d 470	. . 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 2826	. 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 424	1 |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) )

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   i^i cin 3120   (/)c0 3414   {csn 3583   class class class wbr 3989   Oncon0 4348   X. cxp 4609   1oc1o 6388   ~~ cen 6716

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-1o 6395  df-en 6719

This theorem is referenced by: (None)