Theorem endisj 5496

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 groups:   x,y,A   x,B,y

Proof of Theorem endisj

Step	Hyp	Ref	Expression
1		endisj.1	. . . 4 |- A e. _V
2		0ex 3446	. . . 4 |- (/) e. _V
3	1, 2	xpsnen 5494	. . 3 |- (A X. {(/)}) ~~ A
4		endisj.2	. . . 4 |- B e. _V
5		1on 5182	. . . . 5 |- 1o e. On
6	5	elisseti 2301	. . . 4 |- 1o e. _V
7	4, 6	xpsnen 5494	. . 3 |- (B X. {1o}) ~~ B
8	3, 7	pm3.2i 307	. 2 |- ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)
9		xp01disj 5188	. 2 |- ((A X. {(/)}) i^i (B X. {1o})) = (/)
10		p0ex 3495	. . . 4 |- {(/)} e. _V
11	1, 10	xpex 4096	. . 3 |- (A X. {(/)}) e. _V
12		snex 3492	. . . 4 |- {1o} e. _V
13	4, 12	xpex 4096	. . 3 |- (B X. {1o}) e. _V
14		breq1 3341	. . . . 5 |- (x = (A X. {(/)}) -> (x ~~ A <-> (A X. {(/)}) ~~ A))
15		breq1 3341	. . . . 5 |- (y = (B X. {1o}) -> (y ~~ B <-> (B X. {1o}) ~~ B))
16	14, 15	bi2anan9 694	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)))
17		ineq12 2791	. . . . 5 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> (x i^i y) = ((A X. {(/)}) i^i (B X. {1o})))
18	17	eqeq1d 1892	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x i^i y) = (/) <-> ((A X. {(/)}) i^i (B X. {1o})) = (/)))
19	16, 18	anbi12d 690	. . 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	cla42ev 2372	. 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 761	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   i^i cin 2592  (/)c0 2875  {csn 3044   class class class wbr 3338  Oncon0 3657   X. cxp 3984  1oc1o 5172   ~~ cen 5423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-1o 5177  df-en 5427

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