Theorem endisj 3341

Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255.

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 group(s):   x,y,A   x,B,y

Proof of Theorem endisj

Step	Hyp	Ref	Expression
1		endisj.1	. . . 4 |- A e. V
2		0ex 1745	. . . 4 |- (/) e. V
3	1, 2	xpsnen 3339	. . 3 |- (A X. {(/)}) ~~ A
4		endisj.2	. . . 4 |- B e. V
5		p0ex 1885	. . . 4 |- {(/)} e. V
6	4, 5	xpsnen 3339	. . 3 |- (B X. {{(/)}}) ~~ B
7	3, 6	pm3.2i 234	. 2 |- ((A X. {(/)}) ~~ A /\ (B X. {{(/)}}) ~~ B)
8		0nep0 1887	. . 3 |- -. (/) = {(/)}
9		xpsndisj 2655	. . 3 |- (-. (/) = {(/)} -> ((A X. {(/)}) i^i (B X. {{(/)}})) = (/))
10	8, 9	ax-mp 6	. 2 |- ((A X. {(/)}) i^i (B X. {{(/)}})) = (/)
11	1, 5	xpex 2488	. . 3 |- (A X. {(/)}) e. V
12		snex 1859	. . . 4 |- {{(/)}} e. V
13	4, 12	xpex 2488	. . 3 |- (B X. {{(/)}}) e. V
14		breq1 2065	. . . . 5 |- (x = (A X. {(/)}) -> (x ~~ A <-> (A X. {(/)}) ~~ A))
15		breq1 2065	. . . . 5 |- (y = (B X. {{(/)}}) -> (y ~~ B <-> (B X. {{(/)}}) ~~ B))
16	14, 15	bi2anan9 478	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {{(/)}})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. {(/)}) ~~ A /\ (B X. {{(/)}}) ~~ B)))
17		ineq12 1640	. . . . 5 |- ((x = (A X. {(/)}) /\ y = (B X. {{(/)}})) -> (x i^i y) = ((A X. {(/)}) i^i (B X. {{(/)}})))
18	17	cleq1d 1109	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {{(/)}})) -> ((x i^i y) = (/) <-> ((A X. {(/)}) i^i (B X. {{(/)}})) = (/)))
19	16, 18	anbi12d 476	. . 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	cla4e2v 1406	. 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, 10, 20	mp2an 520	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:  -. wn 1   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348   i^i cin 1486  (/)c0 1707  {csn 1808   class class class wbr 2054   X. cxp 2408   ~~ cen 3271

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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