Theorem endisj 4417

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 2701	. . . 4 |- (/) e. V
3	1, 2	xpsnen 4415	. . 3 |- (A X. {(/)}) ~~ A
4		endisj.2	. . . 4 |- B e. V
5		1on 4122	. . . . 5 |- 1o e. On
6	5	elisseti 1809	. . . 4 |- 1o e. V
7	4, 6	xpsnen 4415	. . 3 |- (B X. {1o}) ~~ B
8	3, 7	pm3.2i 285	. 2 |- ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)
9		xp01disj 4127	. 2 |- ((A X. {(/)}) i^i (B X. {1o})) = (/)
10		p0ex 2760	. . . 4 |- {(/)} e. V
11	1, 10	xpex 3250	. . 3 |- (A X. {(/)}) e. V
12		snex 2740	. . . 4 |- {1o} e. V
13	4, 12	xpex 3250	. . 3 |- (B X. {1o}) e. V
14		breq1 2612	. . . . 5 |- (x = (A X. {(/)}) -> (x ~~ A <-> (A X. {(/)}) ~~ A))
15		breq1 2612	. . . . 5 |- (y = (B X. {1o}) -> (y ~~ B <-> (B X. {1o}) ~~ B))
16	14, 15	bi2anan9 630	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)))
17		ineq12 2202	. . . . 5 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> (x i^i y) = ((A X. {(/)}) i^i (B X. {1o})))
18	17	eqeq1d 1475	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x i^i y) = (/) <-> ((A X. {(/)}) i^i (B X. {1o})) = (/)))
19	16, 18	anbi12d 626	. . 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 1861	. 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 695	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609  Oncon0 2938   X. cxp 3158  1oc1o 4112   ~~ cen 4348

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-1o 4117  df-en 4351

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