Theorem unen 4423

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 2869	. . . . 5 |- ((B e. V /\ D e. V) <-> (B u. D) e. V)
2		breng 4366	. . . . . 6 |- (B e. V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
3		breng 4366	. . . . . 6 |- (D e. V -> (C ~~ D <-> E.g g:C-1-1-onto->D))
4	2, 3	bi2anan9 631	. . . . 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 4366	. . . . . . . 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 3701	. . . . . . . . 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 1810	. . . . . . . . . . 11 |- f e. V
9		visset 1810	. . . . . . . . . . 11 |- g e. V
10	8, 9	unex 2868	. . . . . . . . . 10 |- (f u. g) e. V
11		f1oeq1 3679	. . . . . . . . . 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 1866	. . . . . . . . 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 1296	. . . . 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 1322	. . . . 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 2657	. . . 4 |- (-. (B u. D) e. V -> ((A u. C) ~~ (B u. D) <-> (A u. C) ~~ (A u. C)))
22		relen 4363	. . . . . . . 8 |- Rel ~~
23	22	brrelexi 3204	. . . . . . 7 |- (A ~~ B -> A e. V)
24	22	brrelexi 3204	. . . . . . 7 |- (C ~~ D -> C e. V)
25	23, 24	anim12i 333	. . . . . 6 |- ((A ~~ B /\ C ~~ D) -> (A e. V /\ C e. V))
26		unexb 2869	. . . . . 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 4380	. . . . 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 955   e. wcel 957  E.wex 979  Vcvv 1808   u. cun 2042   i^i cin 2043  (/)c0 2277   class class class wbr 2615  -1-1-onto->wf1o 3177   ~~ cen 4357

This theorem is referenced by:  undom 4427  limensuci 4495  phplem2 4498  pssnn 4522  unfi 4537  pm54.43 4555  infensuc 4621  cdaun 4905  cdaen 4907  cda1en 4909  cdacomen 4912  cdaassen 4913  xpcdaen 4914

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-en 4360

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