Theorem endisj 6052

Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by set.mm contributors, 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 4111	. . . . 5 |- (/) e. _V
3	2	complex 4105	. . . 4 |- ∼ (/) e. _V
4	1, 3	xpsnen 6050	. . 3 |- (A X. { ∼ (/)}) ~~ A
5		endisj.2	. . . 4 |- B e. _V
6	5, 2	xpsnen 6050	. . 3 |- (B X. {(/)}) ~~ B
7	4, 6	pm3.2i 441	. 2 |- ((A X. { ∼ (/)}) ~~ A /\ (B X. {(/)}) ~~ B)
8		necompl 3545	. . 3 |- ∼ (/) =/= (/)
9	3, 8	xpnedisj 5514	. 2 |- ((A X. { ∼ (/)}) i^i (B X. {(/)})) = (/)
10		snex 4112	. . . 4 |- { ∼ (/)} e. _V
11	1, 10	xpex 5116	. . 3 |- (A X. { ∼ (/)}) e. _V
12		snex 4112	. . . 4 |- {(/)} e. _V
13	5, 12	xpex 5116	. . 3 |- (B X. {(/)}) e. _V
14		breq1 4643	. . . . 5 |- (x = (A X. { ∼ (/)}) -> (x ~~ A <-> (A X. { ∼ (/)}) ~~ A))
15		breq1 4643	. . . . 5 |- (y = (B X. {(/)}) -> (y ~~ B <-> (B X. {(/)}) ~~ B))
16	14, 15	bi2anan9 843	. . . 4 |- ((x = (A X. { ∼ (/)}) /\ y = (B X. {(/)})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. { ∼ (/)}) ~~ A /\ (B X. {(/)}) ~~ B)))
17		ineq12 3453	. . . . 5 |- ((x = (A X. { ∼ (/)}) /\ y = (B X. {(/)})) -> (x i^i y) = ((A X. { ∼ (/)}) i^i (B X. {(/)})))
18	17	eqeq1d 2361	. . . 4 |- ((x = (A X. { ∼ (/)}) /\ y = (B X. {(/)})) -> ((x i^i y) = (/) <-> ((A X. { ∼ (/)}) i^i (B X. {(/)})) = (/)))
19	16, 18	anbi12d 691	. . 3 |- ((x = (A X. { ∼ (/)}) /\ y = (B X. {(/)})) -> (((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/)) <-> (((A X. { ∼ (/)}) ~~ A /\ (B X. {(/)}) ~~ B) /\ ((A X. { ∼ (/)}) i^i (B X. {(/)})) = (/))))
20	11, 13, 19	spc2ev 2948	. 2 |- ((((A X. { ∼ (/)}) ~~ A /\ (B X. {(/)}) ~~ B) /\ ((A X. { ∼ (/)}) i^i (B X. {(/)})) = (/)) -> E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/)))
21	7, 9, 20	mp2an 653	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   ∼ ccompl 3206   i^i cin 3209  (/)c0 3551  {csn 3738   class class class wbr 4640   X. cxp 4771   ~~ cen 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-en 6030

This theorem is referenced by: (None)