Theorem endisj 6469

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 3931	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 6466	. . 3 |- ( A X. { (/) } ) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 6119	. . . . 5 |- 1o e. On
6	5	elexi 2622	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 6466	. . 3 |- ( B X. { 1o } ) ~~ B
8	3, 7	pm3.2i 266	. 2 |- ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B )
9		xp01disj 6129	. 2 |- ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/)
10		p0ex 3986	. . . 4 |- { (/) } e. _V
11	1, 10	xpex 4510	. . 3 |- ( A X. { (/) } ) e. _V
12	6	snex 3983	. . . 4 |- { 1o } e. _V
13	4, 12	xpex 4510	. . 3 |- ( B X. { 1o } ) e. _V
14		breq1 3814	. . . . 5 |- ( x = ( A X. { (/) } ) -> ( x ~~ A <-> ( A X. { (/) } ) ~~ A ) )
15		breq1 3814	. . . . 5 |- ( y = ( B X. { 1o } ) -> ( y ~~ B <-> ( B X. { 1o } ) ~~ B ) )
16	14, 15	bi2anan9 571	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x ~~ A /\ y ~~ B ) <-> ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) ) )
17		ineq12 3180	. . . . 5 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( x i^i y ) = ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) )
18	17	eqeq1d 2091	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x i^i y ) = (/) <-> ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) )
19	16, 18	anbi12d 457	. . 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 2704	. 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 417	1 |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) )

Syntax hints:   /\ wa 102   = wceq 1285   E.wex 1422   e. wcel 1434   _Vcvv 2612   i^i cin 2983   (/)c0 3269   {csn 3422   class class class wbr 3811   Oncon0 4153   X. cxp 4398   1oc1o 6105   ~~ cen 6384

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-1o 6112  df-en 6387

This theorem is referenced by: (None)