Theorem infcda 9030

Description: The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infcda	|- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A +c B ) ~~ ( A u. B ) )

Proof of Theorem infcda

Step	Hyp	Ref	Expression
1		unnum 9022	. . . . . . 7 |- ( ( A e. dom card /\ B e. dom card ) -> ( A u. B ) e. dom card )
2	1	3adant3 1081	. . . . . 6 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A u. B ) e. dom card )
3		ssun2 3777	. . . . . 6 |- B C_ ( A u. B )
4		ssdomg 8001	. . . . . 6 |- ( ( A u. B ) e. dom card -> ( B C_ ( A u. B ) -> B ~<_ ( A u. B ) ) )
5	2, 3, 4	mpisyl 21	. . . . 5 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> B ~<_ ( A u. B ) )
6		cdadom2 9009	. . . . 5 |- ( B ~<_ ( A u. B ) -> ( A +c B ) ~<_ ( A +c ( A u. B ) ) )
7	5, 6	syl 17	. . . 4 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A +c B ) ~<_ ( A +c ( A u. B ) ) )
8		cdacomen 9003	. . . 4 |- ( A +c ( A u. B ) ) ~~ ( ( A u. B ) +c A )
9		domentr 8015	. . . 4 |- ( ( ( A +c B ) ~<_ ( A +c ( A u. B ) ) /\ ( A +c ( A u. B ) ) ~~ ( ( A u. B ) +c A ) ) -> ( A +c B ) ~<_ ( ( A u. B ) +c A ) )
10	7, 8, 9	sylancl 694	. . 3 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A +c B ) ~<_ ( ( A u. B ) +c A ) )
11		simp3 1063	. . . . 5 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> om ~<_ A )
12		ssun1 3776	. . . . . 6 |- A C_ ( A u. B )
13		ssdomg 8001	. . . . . 6 |- ( ( A u. B ) e. dom card -> ( A C_ ( A u. B ) -> A ~<_ ( A u. B ) ) )
14	2, 12, 13	mpisyl 21	. . . . 5 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> A ~<_ ( A u. B ) )
15		domtr 8009	. . . . 5 |- ( ( om ~<_ A /\ A ~<_ ( A u. B ) ) -> om ~<_ ( A u. B ) )
16	11, 14, 15	syl2anc 693	. . . 4 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> om ~<_ ( A u. B ) )
17		infcdaabs 9028	. . . 4 |- ( ( ( A u. B ) e. dom card /\ om ~<_ ( A u. B ) /\ A ~<_ ( A u. B ) ) -> ( ( A u. B ) +c A ) ~~ ( A u. B ) )
18	2, 16, 14, 17	syl3anc 1326	. . 3 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( ( A u. B ) +c A ) ~~ ( A u. B ) )
19		domentr 8015	. . 3 |- ( ( ( A +c B ) ~<_ ( ( A u. B ) +c A ) /\ ( ( A u. B ) +c A ) ~~ ( A u. B ) ) -> ( A +c B ) ~<_ ( A u. B ) )
20	10, 18, 19	syl2anc 693	. 2 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A +c B ) ~<_ ( A u. B ) )
21		uncdadom 8993	. . 3 |- ( ( A e. dom card /\ B e. dom card ) -> ( A u. B ) ~<_ ( A +c B ) )
22	21	3adant3 1081	. 2 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A u. B ) ~<_ ( A +c B ) )
23		sbth 8080	. 2 |- ( ( ( A +c B ) ~<_ ( A u. B ) /\ ( A u. B ) ~<_ ( A +c B ) ) -> ( A +c B ) ~~ ( A u. B ) )
24	20, 22, 23	syl2anc 693	1 |- ( ( A e. dom card /\ B e. dom card /\ om ~<_ A ) -> ( A +c B ) ~~ ( A u. B ) )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   u. cun 3572   C_ wss 3574   class class class wbr 4653   dom cdm 5114  (class class class)co 6650   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   cardccrd 8761   +c ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  alephadd  9399