Theorem endisj 4524

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 2762	. . . 4 |- (/) e. V
3	1, 2	xpsnen 4522	. . 3 |- (A X. {(/)}) ~~ A
4		endisj.2	. . . 4 |- B e. V
5		1on 4222	. . . . 5 |- 1o e. On
6	5	elisseti 1856	. . . 4 |- 1o e. V
7	4, 6	xpsnen 4522	. . 3 |- (B X. {1o}) ~~ B
8	3, 7	pm3.2i 283	. 2 |- ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)
9		xp01disj 4227	. 2 |- ((A X. {(/)}) i^i (B X. {1o})) = (/)
10		p0ex 2823	. . . 4 |- {(/)} e. V
11	1, 10	xpex 3322	. . 3 |- (A X. {(/)}) e. V
12		snex 2802	. . . 4 |- {1o} e. V
13	4, 12	xpex 3322	. . 3 |- (B X. {1o}) e. V
14		breq1 2672	. . . . 5 |- (x = (A X. {(/)}) -> (x ~~ A <-> (A X. {(/)}) ~~ A))
15		breq1 2672	. . . . 5 |- (y = (B X. {1o}) -> (y ~~ B <-> (B X. {1o}) ~~ B))
16	14, 15	bi2anan9 634	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)))
17		ineq12 2256	. . . . 5 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> (x i^i y) = ((A X. {(/)}) i^i (B X. {1o})))
18	17	eqeq1d 1520	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x i^i y) = (/) <-> ((A X. {(/)}) i^i (B X. {1o})) = (/)))
19	16, 18	anbi12d 630	. . 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 1908	. 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 700	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 221   = wceq 988   e. wcel 990  E.wex 1012  Vcvv 1849   i^i cin 2090  (/)c0 2324  {csn 2454   class class class wbr 2669  Oncon0 3003   X. cxp 3223  1oc1o 4212   ~~ cen 4451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 779  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-tp 2460  df-op 2461  df-uni 2552  df-int 2582  df-br 2670  df-opab 2718  df-tr 2732  df-eprel 2886  df-id 2889  df-po 2894  df-so 2904  df-fr 2972  df-we 2989  df-ord 3006  df-on 3007  df-suc 3009  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-1o 4217  df-en 4455

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