Theorem unen 3338

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.

Assertion

Ref	Expression
unen	|- (((A ~~ B /\ C ~~ D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (A u. C) ~~ (B u. D))

Proof of Theorem unen

Step	Hyp	Ref	Expression
1		unexb 1950	. . . . 5 |- ((B e. V /\ D e. V) <-> (B u. D) e. V)
2		breng 3280	. . . . . 6 |- (B e. V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
3		breng 3280	. . . . . 6 |- (D e. V -> (C ~~ D <-> E.g g:C-1-1-onto->D))
4	2, 3	bi2anan9 478	. . . . 5 |- ((B e. V /\ D e. V) -> ((A ~~ B /\ C ~~ D) <-> (E.f f:A-1-1-onto->B /\ E.g g:C-1-1-onto->D)))
5	1, 4	sylbir 176	. . . 4 |- ((B u. D) e. V -> ((A ~~ B /\ C ~~ D) <-> (E.f f:A-1-1-onto->B /\ E.g g:C-1-1-onto->D)))
6		breng 3280	. . . . . . . 8 |- ((B u. D) e. V -> ((A u. C) ~~ (B u. D) <-> E.h h:(A u. C)-1-1-onto->(B u. D)))
7		f1oun 2815	. . . . . . . . 9 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (f u. g):(A u. C)-1-1-onto->(B u. D))
8		visset 1350	. . . . . . . . . . 11 |- f e. V
9		visset 1350	. . . . . . . . . . 11 |- g e. V
10	8, 9	unex 1949	. . . . . . . . . 10 |- (f u. g) e. V
11		f1oeq1 2795	. . . . . . . . . 10 |- (h = (f u. g) -> (h:(A u. C)-1-1-onto->(B u. D) <-> (f u. g):(A u. C)-1-1-onto->(B u. D)))
12	10, 11	cla4ev 1401	. . . . . . . . 9 |- ((f u. g):(A u. C)-1-1-onto->(B u. D) -> E.h h:(A u. C)-1-1-onto->(B u. D))
13	7, 12	syl 12	. . . . . . . 8 |- (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> E.h h:(A u. C)-1-1-onto->(B u. D))
14	6, 13	syl5bir 184	. . . . . . 7 |- ((B u. D) e. V -> (((f:A-1-1-onto->B /\ g:C-1-1-onto->D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (A u. C) ~~ (B u. D)))
15	14	exp3a 292	. . . . . 6 |- ((B u. D) e. V -> ((f:A-1-1-onto->B /\ g:C-1-1-onto->D) -> (((A i^i C) = (/) /\ (B i^i D) = (/)) -> (A u. C) ~~ (B u. D))))
16	15	19.23advv 955	. . . . 5 |- ((B u. D) e. V -> (E.fE.g(f:A-1-1-onto->B /\ g:C-1-1-onto->D) -> (((A i^i C) = (/) /\ (B i^i D) = (/)) -> (A u. C) ~~ (B u. D))))
17		eeanv 980	. . . . 5 |- (E.fE.g(f:A-1-1-onto->B /\ g:C-1-1-onto->D) <-> (E.f f:A-1-1-onto->B /\ E.g g:C-1-1-onto->D))
18	16, 17	syl5ibr 182	. . . 4 |- ((B u. D) e. V -> ((E.f f:A-1-1-onto->B /\ E.g g:C-1-1-onto->D) -> (((A i^i C) = (/) /\ (B i^i D) = (/)) -> (A u. C) ~~ (B u. D))))
19	5, 18	sylbid 178	. . 3 |- ((B u. D) e. V -> ((A ~~ B /\ C ~~ D) -> (((A i^i C) = (/) /\ (B i^i D) = (/)) -> (A u. C) ~~ (B u. D))))
20	19	imp3a 279	. 2 |- ((B u. D) e. V -> (((A ~~ B /\ C ~~ D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (A u. C) ~~ (B u. D)))
21		brprc 2097	. . . 4 |- (-. (B u. D) e. V -> ((A u. C) ~~ (B u. D) <-> (A u. C) ~~ (A u. C)))
22		relen 3277	. . . . . . . 8 |- Rel ~~
23	22	brrelexi 2447	. . . . . . 7 |- (A ~~ B -> A e. V)
24	22	brrelexi 2447	. . . . . . 7 |- (C ~~ D -> C e. V)
25	23, 24	anim12i 268	. . . . . 6 |- ((A ~~ B /\ C ~~ D) -> (A e. V /\ C e. V))
26		unexb 1950	. . . . . 6 |- ((A e. V /\ C e. V) <-> (A u. C) e. V)
27	25, 26	sylib 173	. . . . 5 |- ((A ~~ B /\ C ~~ D) -> (A u. C) e. V)
28		enrefg 3294	. . . . 5 |- ((A u. C) e. V -> (A u. C) ~~ (A u. C))
29	27, 28	syl 12	. . . 4 |- ((A ~~ B /\ C ~~ D) -> (A u. C) ~~ (A u. C))
30	21, 29	syl5bir 184	. . 3 |- (-. (B u. D) e. V -> ((A ~~ B /\ C ~~ D) -> (A u. C) ~~ (B u. D)))
31	30	adantrd 308	. 2 |- (-. (B u. D) e. V -> (((A ~~ B /\ C ~~ D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (A u. C) ~~ (B u. D)))
32	20, 31	pm2.61i 110	1 |- (((A ~~ B /\ C ~~ D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (A u. C) ~~ (B u. D))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   i^i cin 1486  (/)c0 1707   class class class wbr 2054  -1-1-onto->wf1o 2421   ~~ cen 3271

This theorem is referenced by:  undom 3342  limensuci 3401  phplem3 3405  pssnn 3428  unfi 3441  infensuc 3484  cdaen 3719  cda1en 3721  cdacomen 3724  cdaassen 3725  xpcdaen 3726

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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