Theorem alephsucdom 8902

Description: A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
alephsucdom	|- ( B e. On -> ( A ~<_ ( aleph ` B ) <-> A ~< ( aleph ` suc B ) ) )

Proof of Theorem alephsucdom

Step	Hyp	Ref	Expression
1		alephordilem1 8896	. . 3 |- ( B e. On -> ( aleph ` B ) ~< ( aleph ` suc B ) )
2		domsdomtr 8095	. . . 4 |- ( ( A ~<_ ( aleph ` B ) /\ ( aleph ` B ) ~< ( aleph ` suc B ) ) -> A ~< ( aleph ` suc B ) )
3	2	ex 450	. . 3 |- ( A ~<_ ( aleph ` B ) -> ( ( aleph ` B ) ~< ( aleph ` suc B ) -> A ~< ( aleph ` suc B ) ) )
4	1, 3	syl5com 31	. 2 |- ( B e. On -> ( A ~<_ ( aleph ` B ) -> A ~< ( aleph ` suc B ) ) )
5		sdomdom 7983	. . . . 5 |- ( A ~< ( aleph ` suc B ) -> A ~<_ ( aleph ` suc B ) )
6		alephon 8892	. . . . . 6 |- ( aleph ` suc B ) e. On
7		ondomen 8860	. . . . . 6 |- ( ( ( aleph ` suc B ) e. On /\ A ~<_ ( aleph ` suc B ) ) -> A e. dom card )
8	6, 7	mpan 706	. . . . 5 |- ( A ~<_ ( aleph ` suc B ) -> A e. dom card )
9		cardid2 8779	. . . . 5 |- ( A e. dom card -> ( card ` A ) ~~ A )
10	5, 8, 9	3syl 18	. . . 4 |- ( A ~< ( aleph ` suc B ) -> ( card ` A ) ~~ A )
11	10	ensymd 8007	. . 3 |- ( A ~< ( aleph ` suc B ) -> A ~~ ( card ` A ) )
12		alephnbtwn2 8895	. . . . . 6 |- -. ( ( aleph ` B ) ~< ( card ` A ) /\ ( card ` A ) ~< ( aleph ` suc B ) )
13	12	imnani 439	. . . . 5 |- ( ( aleph ` B ) ~< ( card ` A ) -> -. ( card ` A ) ~< ( aleph ` suc B ) )
14		ensdomtr 8096	. . . . . 6 |- ( ( ( card ` A ) ~~ A /\ A ~< ( aleph ` suc B ) ) -> ( card ` A ) ~< ( aleph ` suc B ) )
15	10, 14	mpancom 703	. . . . 5 |- ( A ~< ( aleph ` suc B ) -> ( card ` A ) ~< ( aleph ` suc B ) )
16	13, 15	nsyl3 133	. . . 4 |- ( A ~< ( aleph ` suc B ) -> -. ( aleph ` B ) ~< ( card ` A ) )
17		cardon 8770	. . . . 5 |- ( card ` A ) e. On
18		alephon 8892	. . . . 5 |- ( aleph ` B ) e. On
19		domtriord 8106	. . . . 5 |- ( ( ( card ` A ) e. On /\ ( aleph ` B ) e. On ) -> ( ( card ` A ) ~<_ ( aleph ` B ) <-> -. ( aleph ` B ) ~< ( card ` A ) ) )
20	17, 18, 19	mp2an 708	. . . 4 |- ( ( card ` A ) ~<_ ( aleph ` B ) <-> -. ( aleph ` B ) ~< ( card ` A ) )
21	16, 20	sylibr 224	. . 3 |- ( A ~< ( aleph ` suc B ) -> ( card ` A ) ~<_ ( aleph ` B ) )
22		endomtr 8014	. . 3 |- ( ( A ~~ ( card ` A ) /\ ( card ` A ) ~<_ ( aleph ` B ) ) -> A ~<_ ( aleph ` B ) )
23	11, 21, 22	syl2anc 693	. 2 |- ( A ~< ( aleph ` suc B ) -> A ~<_ ( aleph ` B ) )
24	4, 23	impbid1 215	1 |- ( B e. On -> ( A ~<_ ( aleph ` B ) <-> A ~< ( aleph ` suc B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   e. wcel 1990   class class class wbr 4653   dom cdm 5114   Oncon0 5723   suc csuc 5725   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   alephcale 8762

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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  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

This theorem is referenced by:  alephsuc2  8903  alephreg  9404