Theorem unen 4575

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 3096	. . . . 5 |- ((B e. V /\ D e. V) <-> (B u. D) e. V)
2		breng 4516	. . . . . 6 |- (B e. V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
3		breng 4516	. . . . . 6 |- (D e. V -> (C ~~ D <-> E.g g:C-1-1-onto->D))
4	2, 3	bi2anan9 635	. . . . 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 199	. . . 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 4516	. . . . . . . 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 3815	. . . . . . . . 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 1859	. . . . . . . . . . 11 |- f e. V
9		visset 1859	. . . . . . . . . . 11 |- g e. V
10	8, 9	unex 3095	. . . . . . . . . 10 |- (f u. g) e. V
11		f1oeq1 3792	. . . . . . . . . 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 1915	. . . . . . . . 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 10	. . . . . . . 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 208	. . . . . . 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 374	. . . . . 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 1335	. . . . 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 1361	. . . . 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 205	. . . 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 201	. . 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 359	. 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 2734	. . . 4 |- (-. (B u. D) e. V -> ((A u. C) ~~ (B u. D) <-> (A u. C) ~~ (A u. C)))
22		relen 4513	. . . . . . . 8 |- Rel ~~
23	22	brrelexi 3291	. . . . . . 7 |- (A ~~ B -> A e. V)
24	22	brrelexi 3291	. . . . . . 7 |- (C ~~ D -> C e. V)
25	23, 24	anim12i 331	. . . . . 6 |- ((A ~~ B /\ C ~~ D) -> (A e. V /\ C e. V))
26		unexb 3096	. . . . . 6 |- ((A e. V /\ C e. V) <-> (A u. C) e. V)
27	25, 26	sylib 196	. . . . 5 |- ((A ~~ B /\ C ~~ D) -> (A u. C) e. V)
28		enrefg 4531	. . . . 5 |- ((A u. C) e. V -> (A u. C) ~~ (A u. C))
29	27, 28	syl 10	. . . 4 |- ((A ~~ B /\ C ~~ D) -> (A u. C) ~~ (A u. C))
30	21, 29	syl5bir 208	. . 3 |- (-. (B u. D) e. V -> ((A ~~ B /\ C ~~ D) -> (A u. C) ~~ (B u. D)))
31	30	adantrd 391	. 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 124	1 |- (((A ~~ B /\ C ~~ D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (A u. C) ~~ (B u. D))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  E.wex 1016  Vcvv 1857   u. cun 2097   i^i cin 2098  (/)c0 2332   class class class wbr 2692  -1-1-onto->wf1o 3262   ~~ cen 4505

This theorem is referenced by:  undom 4579  limensuci 4653  phplem2 4656  pssnn 4681  unfi 4697  pm54.43 4715  infensuc 4784  unsnen 4983  cdaun 5072  cdaen 5075  cda1en 5078  cdacomen 5081  cdaassen 5082  xpcdaen 5083

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-en 4509

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