Theorem unen 4434

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 2873	. . . . 5 |- ((B e. V /\ D e. V) <-> (B u. D) e. V)
2		breng 4375	. . . . . 6 |- (B e. V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
3		breng 4375	. . . . . 6 |- (D e. V -> (C ~~ D <-> E.g g:C-1-1-onto->D))
4	2, 3	bi2anan9 632	. . . . 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 4375	. . . . . . . 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 3706	. . . . . . . . 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 1813	. . . . . . . . . . 11 |- f e. V
9		visset 1813	. . . . . . . . . . 11 |- g e. V
10	8, 9	unex 2872	. . . . . . . . . 10 |- (f u. g) e. V
11		f1oeq1 3684	. . . . . . . . . 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 1869	. . . . . . . . 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 1297	. . . . 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 1323	. . . . 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 2661	. . . 4 |- (-. (B u. D) e. V -> ((A u. C) ~~ (B u. D) <-> (A u. C) ~~ (A u. C)))
22		relen 4372	. . . . . . . 8 |- Rel ~~
23	22	brrelexi 3208	. . . . . . 7 |- (A ~~ B -> A e. V)
24	22	brrelexi 3208	. . . . . . 7 |- (C ~~ D -> C e. V)
25	23, 24	anim12i 333	. . . . . 6 |- ((A ~~ B /\ C ~~ D) -> (A e. V /\ C e. V))
26		unexb 2873	. . . . . 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 4390	. . . . 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 956   e. wcel 958  E.wex 980  Vcvv 1811   u. cun 2045   i^i cin 2046  (/)c0 2280   class class class wbr 2619  -1-1-onto->wf1o 3181   ~~ cen 4364

This theorem is referenced by:  undom 4438  limensuci 4506  phplem2 4509  pssnn 4534  unfi 4551  unfiOLD 4552  pm54.43 4572  infensuc 4638  cdaun 4922  cdaen 4924  cda1en 4926  cdacomen 4929  cdaassen 4930  xpcdaen 4931

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368

Copyright terms: Public domain