Theorem unen 5493

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 3797	. . . . 5 |- ((B e. _V /\ D e. _V) <-> (B u. D) e. _V)
2		breng 5434	. . . . . 6 |- (B e. _V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
3		breng 5434	. . . . . 6 |- (D e. _V -> (C ~~ D <-> E.g g:C-1-1-onto->D))
4	2, 3	bi2anan9 694	. . . . 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 218	. . . 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 5434	. . . . . . . 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 4658	. . . . . . . . 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 2295	. . . . . . . . . . 11 |- f e. _V
9		visset 2295	. . . . . . . . . . 11 |- g e. _V
10	8, 9	unex 3796	. . . . . . . . . 10 |- (f u. g) e. _V
11		f1oeq1 4630	. . . . . . . . . 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 2371	. . . . . . . . 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 227	. . . . . . 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 405	. . . . . 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 1676	. . . . 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 1707	. . . . 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 224	. . . 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 220	. . 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 388	. 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 3386	. . . 4 |- (-. (B u. D) e. _V -> ((A u. C) ~~ (B u. D) <-> (A u. C) ~~ (A u. C)))
22		relen 5431	. . . . . . . 8 |- Rel ~~
23	22	brrelexi 4029	. . . . . . 7 |- (A ~~ B -> A e. _V)
24	22	brrelexi 4029	. . . . . . 7 |- (C ~~ D -> C e. _V)
25	23, 24	anim12i 360	. . . . . 6 |- ((A ~~ B /\ C ~~ D) -> (A e. _V /\ C e. _V))
26		unexb 3797	. . . . . 6 |- ((A e. _V /\ C e. _V) <-> (A u. C) e. _V)
27	25, 26	sylib 215	. . . . 5 |- ((A ~~ B /\ C ~~ D) -> (A u. C) e. _V)
28		enrefg 5449	. . . . 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 227	. . 3 |- (-. (B u. D) e. _V -> ((A ~~ B /\ C ~~ D) -> (A u. C) ~~ (B u. D)))
31	30	adantrd 427	. 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 140	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   u. cun 2591   i^i cin 2592  (/)c0 2875   class class class wbr 3338  -1-1-onto->wf1o 3997   ~~ cen 5423

This theorem is referenced by:  undom 5497  limensuci 5600  phplem2 5603  pssnn 5628  unfi 5644  pm54.43 5662  infensuc 5745  unsnen 5985  cdaun 6070  cdaen 6073  cda1en 6076  cdacomen 6079  cdaassen 6080  xpcdaen 6081  dif1en 10172

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-en 5427

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