Theorem endisj 4500

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 2766	. . . 4 |- (/) e. V
3	1, 2	xpsnen 4498	. . 3 |- (A X. {(/)}) ~~ A
4		endisj.2	. . . 4 |- B e. V
5		1on 4196	. . . . 5 |- 1o e. On
6	5	elisseti 1865	. . . 4 |- 1o e. V
7	4, 6	xpsnen 4498	. . 3 |- (B X. {1o}) ~~ B
8	3, 7	pm3.2i 292	. 2 |- ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)
9		xp01disj 4201	. 2 |- ((A X. {(/)}) i^i (B X. {1o})) = (/)
10		p0ex 2826	. . . 4 |- {(/)} e. V
11	1, 10	xpex 3317	. . 3 |- (A X. {(/)}) e. V
12		snex 2806	. . . 4 |- {1o} e. V
13	4, 12	xpex 3317	. . 3 |- (B X. {1o}) e. V
14		breq1 2677	. . . . 5 |- (x = (A X. {(/)}) -> (x ~~ A <-> (A X. {(/)}) ~~ A))
15		breq1 2677	. . . . 5 |- (y = (B X. {1o}) -> (y ~~ B <-> (B X. {1o}) ~~ B))
16	14, 15	bi2anan9 643	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)))
17		ineq12 2263	. . . . 5 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> (x i^i y) = ((A X. {(/)}) i^i (B X. {1o})))
18	17	eqeq1d 1530	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x i^i y) = (/) <-> ((A X. {(/)}) i^i (B X. {1o})) = (/)))
19	16, 18	anbi12d 639	. . 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 1917	. 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 709	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 230   = wceq 997   e. wcel 999  E.wex 1021  Vcvv 1858   i^i cin 2097  (/)c0 2331  {csn 2461   class class class wbr 2674  Oncon0 3005   X. cxp 3225  1oc1o 4186   ~~ cen 4425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-suc 3011  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-f1 3252  df-fo 3253  df-f1o 3254  df-1o 4191  df-en 4429

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