Theorem endisj 6234

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.xE.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 3875	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 6231	. . 3 |- (A X. { (/) }) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 5947	. . . . 5 |- 1o e. On
6	5	elexi 2561	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 6231	. . 3 |- (B X. { 1o }) ~~ B
8	3, 7	pm3.2i 257	. 2 |- ((A X. { (/) }) ~~ A /\ (B X. { 1o }) ~~ B)
9		xp01disj 5956	. 2 |- ((A X. { (/) }) i^i (B X. { 1o })) = (/)
10		p0ex 3930	. . . 4 |- { (/) } e. _V
11	1, 10	xpex 4396	. . 3 |- (A X. { (/) }) e. _V
12	6	snex 3928	. . . 4 |- { 1o } e. _V
13	4, 12	xpex 4396	. . 3 |- (B X. { 1o }) e. _V
14		breq1 3758	. . . . 5 |- (x = (A X. { (/) }) -> (x ~~ A <-> (A X. { (/) }) ~~ A))
15		breq1 3758	. . . . 5 |- (y = (B X. { 1o }) -> (y ~~ B <-> (B X. { 1o }) ~~ B))
16	14, 15	bi2anan9 538	. . . 4 |- ((x = (A X. { (/) }) /\ y = (B X. { 1o })) -> ((x ~~ A /\ y ~~ B) <-> ((A X. { (/) }) ~~ A /\ (B X. { 1o }) ~~ B)))
17		ineq12 3127	. . . . 5 |- ((x = (A X. { (/) }) /\ y = (B X. { 1o })) -> (x i^i y) = ((A X. { (/) }) i^i (B X. { 1o })))
18	17	eqeq1d 2045	. . . 4 |- ((x = (A X. { (/) }) /\ y = (B X. { 1o })) -> ((x i^i y) = (/) <-> ((A X. { (/) }) i^i (B X. { 1o })) = (/)))
19	16, 18	anbi12d 442	. . 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 2642	. 2 |- ((((A X. { (/) }) ~~ A /\ (B X. { 1o }) ~~ B) /\ ((A X. { (/) }) i^i (B X. { 1o })) = (/)) -> E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/)))
21	8, 9, 20	mp2an 402	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551   i^i cin 2910   (/)c0 3218   {csn 3367   class class class wbr 3755   Oncon0 4066   X. cxp 4286   1oc1o 5933   ~~ cen 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-1o 5940  df-en 6158

This theorem is referenced by: (None)