Theorem alephadd 9399

Description: The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
alephadd	|- ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) )

Proof of Theorem alephadd

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( ( aleph ` A ) +c ( aleph ` B ) ) e. _V
2		alephfnon 8888	. . . . . . . 8 |- aleph Fn On
3		fndm 5990	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
4	2, 3	ax-mp 5	. . . . . . 7 |- dom aleph = On
5	4	eleq2i 2693	. . . . . 6 |- ( A e. dom aleph <-> A e. On )
6	5	notbii 310	. . . . 5 |- ( -. A e. dom aleph <-> -. A e. On )
7	4	eleq2i 2693	. . . . . 6 |- ( B e. dom aleph <-> B e. On )
8	7	notbii 310	. . . . 5 |- ( -. B e. dom aleph <-> -. B e. On )
9		0ex 4790	. . . . . . . 8 |- (/) e. _V
10		cdaval 8992	. . . . . . . 8 |- ( ( (/) e. _V /\ (/) e. _V ) -> ( (/) +c (/) ) = ( ( (/) X. { (/) } ) u. ( (/) X. { 1o } ) ) )
11	9, 9, 10	mp2an 708	. . . . . . 7 |- ( (/) +c (/) ) = ( ( (/) X. { (/) } ) u. ( (/) X. { 1o } ) )
12		xpundi 5171	. . . . . . 7 |- ( (/) X. ( { (/) } u. { 1o } ) ) = ( ( (/) X. { (/) } ) u. ( (/) X. { 1o } ) )
13		0xp 5199	. . . . . . 7 |- ( (/) X. ( { (/) } u. { 1o } ) ) = (/)
14	11, 12, 13	3eqtr2i 2650	. . . . . 6 |- ( (/) +c (/) ) = (/)
15		ndmfv 6218	. . . . . . 7 |- ( -. A e. dom aleph -> ( aleph ` A ) = (/) )
16		ndmfv 6218	. . . . . . 7 |- ( -. B e. dom aleph -> ( aleph ` B ) = (/) )
17	15, 16	oveqan12d 6669	. . . . . 6 |- ( ( -. A e. dom aleph /\ -. B e. dom aleph ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) = ( (/) +c (/) ) )
18	15	adantr 481	. . . . . . . 8 |- ( ( -. A e. dom aleph /\ -. B e. dom aleph ) -> ( aleph ` A ) = (/) )
19	16	adantl 482	. . . . . . . 8 |- ( ( -. A e. dom aleph /\ -. B e. dom aleph ) -> ( aleph ` B ) = (/) )
20	18, 19	uneq12d 3768	. . . . . . 7 |- ( ( -. A e. dom aleph /\ -. B e. dom aleph ) -> ( ( aleph ` A ) u. ( aleph ` B ) ) = ( (/) u. (/) ) )
21		un0 3967	. . . . . . 7 |- ( (/) u. (/) ) = (/)
22	20, 21	syl6eq 2672	. . . . . 6 |- ( ( -. A e. dom aleph /\ -. B e. dom aleph ) -> ( ( aleph ` A ) u. ( aleph ` B ) ) = (/) )
23	14, 17, 22	3eqtr4a 2682	. . . . 5 |- ( ( -. A e. dom aleph /\ -. B e. dom aleph ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) = ( ( aleph ` A ) u. ( aleph ` B ) ) )
24	6, 8, 23	syl2anbr 497	. . . 4 |- ( ( -. A e. On /\ -. B e. On ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) = ( ( aleph ` A ) u. ( aleph ` B ) ) )
25		eqeng 7989	. . . 4 |- ( ( ( aleph ` A ) +c ( aleph ` B ) ) e. _V -> ( ( ( aleph ` A ) +c ( aleph ` B ) ) = ( ( aleph ` A ) u. ( aleph ` B ) ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) ) )
26	1, 24, 25	mpsyl 68	. . 3 |- ( ( -. A e. On /\ -. B e. On ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
27	26	ex 450	. 2 |- ( -. A e. On -> ( -. B e. On -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) ) )
28		alephgeom 8905	. . 3 |- ( A e. On <-> om C_ ( aleph ` A ) )
29		fvex 6201	. . . . 5 |- ( aleph ` A ) e. _V
30		ssdomg 8001	. . . . 5 |- ( ( aleph ` A ) e. _V -> ( om C_ ( aleph ` A ) -> om ~<_ ( aleph ` A ) ) )
31	29, 30	ax-mp 5	. . . 4 |- ( om C_ ( aleph ` A ) -> om ~<_ ( aleph ` A ) )
32		alephon 8892	. . . . . 6 |- ( aleph ` A ) e. On
33		onenon 8775	. . . . . 6 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
34	32, 33	ax-mp 5	. . . . 5 |- ( aleph ` A ) e. dom card
35		alephon 8892	. . . . . 6 |- ( aleph ` B ) e. On
36		onenon 8775	. . . . . 6 |- ( ( aleph ` B ) e. On -> ( aleph ` B ) e. dom card )
37	35, 36	ax-mp 5	. . . . 5 |- ( aleph ` B ) e. dom card
38		infcda 9030	. . . . 5 |- ( ( ( aleph ` A ) e. dom card /\ ( aleph ` B ) e. dom card /\ om ~<_ ( aleph ` A ) ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
39	34, 37, 38	mp3an12 1414	. . . 4 |- ( om ~<_ ( aleph ` A ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
40	31, 39	syl 17	. . 3 |- ( om C_ ( aleph ` A ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
41	28, 40	sylbi 207	. 2 |- ( A e. On -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
42		alephgeom 8905	. . 3 |- ( B e. On <-> om C_ ( aleph ` B ) )
43		fvex 6201	. . . . 5 |- ( aleph ` B ) e. _V
44		ssdomg 8001	. . . . 5 |- ( ( aleph ` B ) e. _V -> ( om C_ ( aleph ` B ) -> om ~<_ ( aleph ` B ) ) )
45	43, 44	ax-mp 5	. . . 4 |- ( om C_ ( aleph ` B ) -> om ~<_ ( aleph ` B ) )
46		cdacomen 9003	. . . . . 6 |- ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` B ) +c ( aleph ` A ) )
47		infcda 9030	. . . . . . 7 |- ( ( ( aleph ` B ) e. dom card /\ ( aleph ` A ) e. dom card /\ om ~<_ ( aleph ` B ) ) -> ( ( aleph ` B ) +c ( aleph ` A ) ) ~~ ( ( aleph ` B ) u. ( aleph ` A ) ) )
48	37, 34, 47	mp3an12 1414	. . . . . 6 |- ( om ~<_ ( aleph ` B ) -> ( ( aleph ` B ) +c ( aleph ` A ) ) ~~ ( ( aleph ` B ) u. ( aleph ` A ) ) )
49		entr 8008	. . . . . 6 |- ( ( ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` B ) +c ( aleph ` A ) ) /\ ( ( aleph ` B ) +c ( aleph ` A ) ) ~~ ( ( aleph ` B ) u. ( aleph ` A ) ) ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` B ) u. ( aleph ` A ) ) )
50	46, 48, 49	sylancr 695	. . . . 5 |- ( om ~<_ ( aleph ` B ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` B ) u. ( aleph ` A ) ) )
51		uncom 3757	. . . . 5 |- ( ( aleph ` B ) u. ( aleph ` A ) ) = ( ( aleph ` A ) u. ( aleph ` B ) )
52	50, 51	syl6breq 4694	. . . 4 |- ( om ~<_ ( aleph ` B ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
53	45, 52	syl 17	. . 3 |- ( om C_ ( aleph ` B ) -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
54	42, 53	sylbi 207	. 2 |- ( B e. On -> ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) )
55	27, 41, 54	pm2.61ii 177	1 |- ( ( aleph ` A ) +c ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Oncon0 5723   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   omcom 7065   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   cardccrd 8761   alephcale 8762   +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-har 8463  df-card 8765  df-aleph 8766  df-cda 8990

This theorem is referenced by: (None)