Theorem endisj 4423

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 2706	. . . 4 |- (/) e. V
3	1, 2	xpsnen 4421	. . 3 |- (A X. {(/)}) ~~ A
4		endisj.2	. . . 4 |- B e. V
5		1on 4128	. . . . 5 |- 1o e. On
6	5	elisseti 1814	. . . 4 |- 1o e. V
7	4, 6	xpsnen 4421	. . 3 |- (B X. {1o}) ~~ B
8	3, 7	pm3.2i 285	. 2 |- ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)
9		xp01disj 4133	. 2 |- ((A X. {(/)}) i^i (B X. {1o})) = (/)
10		p0ex 2765	. . . 4 |- {(/)} e. V
11	1, 10	xpex 3255	. . 3 |- (A X. {(/)}) e. V
12		snex 2745	. . . 4 |- {1o} e. V
13	4, 12	xpex 3255	. . 3 |- (B X. {1o}) e. V
14		breq1 2617	. . . . 5 |- (x = (A X. {(/)}) -> (x ~~ A <-> (A X. {(/)}) ~~ A))
15		breq1 2617	. . . . 5 |- (y = (B X. {1o}) -> (y ~~ B <-> (B X. {1o}) ~~ B))
16	14, 15	bi2anan9 631	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x ~~ A /\ y ~~ B) <-> ((A X. {(/)}) ~~ A /\ (B X. {1o}) ~~ B)))
17		ineq12 2208	. . . . 5 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> (x i^i y) = ((A X. {(/)}) i^i (B X. {1o})))
18	17	eqeq1d 1480	. . . 4 |- ((x = (A X. {(/)}) /\ y = (B X. {1o})) -> ((x i^i y) = (/) <-> ((A X. {(/)}) i^i (B X. {1o})) = (/)))
19	16, 18	anbi12d 627	. . 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 1866	. 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 696	1 |- E.xE.y((x ~~ A /\ y ~~ B) /\ (x i^i y) = (/))

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  Vcvv 1807   i^i cin 2042  (/)c0 2276  {csn 2405   class class class wbr 2614  Oncon0 2943   X. cxp 3163  1oc1o 4118   ~~ cen 4354

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-1o 4123  df-en 4357

Copyright terms: Public domain