Theorem unen 4414

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 2864	. . . . 5 |- ((B e. V /\ D e. V) <-> (B u. D) e. V)
2		breng 4357	. . . . . 6 |- (B e. V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
3		breng 4357	. . . . . 6 |- (D e. V -> (C ~~ D <-> E.g g:C-1-1-onto->D))
4	2, 3	bi2anan9 630	. . . . 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 201	. . . 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 4357	. . . . . . . 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 3691	. . . . . . . . 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 1804	. . . . . . . . . . 11 |- f e. V
9		visset 1804	. . . . . . . . . . 11 |- g e. V
10	8, 9	unex 2863	. . . . . . . . . 10 |- (f u. g) e. V
11		f1oeq1 3669	. . . . . . . . . 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 1860	. . . . . . . . 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 210	. . . . . . 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 375	. . . . . 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 1292	. . . . 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 1318	. . . . 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 207	. . . 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 203	. . 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 361	. 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 2651	. . . 4 |- (-. (B u. D) e. V -> ((A u. C) ~~ (B u. D) <-> (A u. C) ~~ (A u. C)))
22		relen 4354	. . . . . . . 8 |- Rel ~~
23	22	brrelexi 3198	. . . . . . 7 |- (A ~~ B -> A e. V)
24	22	brrelexi 3198	. . . . . . 7 |- (C ~~ D -> C e. V)
25	23, 24	anim12i 333	. . . . . 6 |- ((A ~~ B /\ C ~~ D) -> (A e. V /\ C e. V))
26		unexb 2864	. . . . . 6 |- ((A e. V /\ C e. V) <-> (A u. C) e. V)
27	25, 26	sylib 198	. . . . 5 |- ((A ~~ B /\ C ~~ D) -> (A u. C) e. V)
28		enrefg 4371	. . . . 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 210	. . 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 126	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 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270   class class class wbr 2609  -1-1-onto->wf1o 3171   ~~ cen 4348

This theorem is referenced by:  undom 4418  limensuci 4486  phplem2 4489  pssnn 4513  unfi 4528  pm54.43 4546  infensuc 4610  cdaun 4894  cdaen 4896  cda1en 4898  cdacomen 4901  cdaassen 4902  xpcdaen 4903

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-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  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-en 4351

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