Theorem endisj 6878

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 4156	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 6875	. . 3 |- ( A X. { (/) } ) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 6476	. . . . 5 |- 1o e. On
6	5	elexi 2772	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 6875	. . 3 |- ( B X. { 1o } ) ~~ B
8	3, 7	pm3.2i 272	. 2 |- ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B )
9		xp01disj 6486	. 2 |- ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/)
10		p0ex 4217	. . . 4 |- { (/) } e. _V
11	1, 10	xpex 4774	. . 3 |- ( A X. { (/) } ) e. _V
12	6	snex 4214	. . . 4 |- { 1o } e. _V
13	4, 12	xpex 4774	. . 3 |- ( B X. { 1o } ) e. _V
14		breq1 4032	. . . . 5 |- ( x = ( A X. { (/) } ) -> ( x ~~ A <-> ( A X. { (/) } ) ~~ A ) )
15		breq1 4032	. . . . 5 |- ( y = ( B X. { 1o } ) -> ( y ~~ B <-> ( B X. { 1o } ) ~~ B ) )
16	14, 15	bi2anan9 606	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x ~~ A /\ y ~~ B ) <-> ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) ) )
17		ineq12 3355	. . . . 5 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( x i^i y ) = ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) )
18	17	eqeq1d 2202	. . . 4 |- ( ( x = ( A X. { (/) } ) /\ y = ( B X. { 1o } ) ) -> ( ( x i^i y ) = (/) <-> ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) ) )
19	16, 18	anbi12d 473	. . 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 2856	. 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 426	1 |- E. x E. y ( ( x ~~ A /\ y ~~ B ) /\ ( x i^i y ) = (/) )

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   i^i cin 3152   (/)c0 3446   {csn 3618   class class class wbr 4029   Oncon0 4394   X. cxp 4657   1oc1o 6462   ~~ cen 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-1o 6469  df-en 6795

This theorem is referenced by: (None)