Theorem alephdom 8904

Description: Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)

Assertion

Ref	Expression
alephdom	|- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( aleph ` A ) ~<_ ( aleph ` B ) ) )

Proof of Theorem alephdom

Step	Hyp	Ref	Expression
1		onsseleq 5765	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) )
2		alephord 8898	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) ~< ( aleph ` B ) ) )
3		sdomdom 7983	. . . . 5 |- ( ( aleph ` A ) ~< ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) )
4	2, 3	syl6bi 243	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A e. B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
5		fvex 6201	. . . . . . 7 |- ( aleph ` A ) e. _V
6		fveq2 6191	. . . . . . 7 |- ( A = B -> ( aleph ` A ) = ( aleph ` B ) )
7		eqeng 7989	. . . . . . 7 |- ( ( aleph ` A ) e. _V -> ( ( aleph ` A ) = ( aleph ` B ) -> ( aleph ` A ) ~~ ( aleph ` B ) ) )
8	5, 6, 7	mpsyl 68	. . . . . 6 |- ( A = B -> ( aleph ` A ) ~~ ( aleph ` B ) )
9	8	a1i 11	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A = B -> ( aleph ` A ) ~~ ( aleph ` B ) ) )
10		endom 7982	. . . . 5 |- ( ( aleph ` A ) ~~ ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) )
11	9, 10	syl6 35	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A = B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
12	4, 11	jaod 395	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( A e. B \/ A = B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
13	1, 12	sylbid 230	. 2 |- ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
14		eloni 5733	. . . . . 6 |- ( B e. On -> Ord B )
15		eloni 5733	. . . . . 6 |- ( A e. On -> Ord A )
16		ordtri2or 5822	. . . . . 6 |- ( ( Ord B /\ Ord A ) -> ( B e. A \/ A C_ B ) )
17	14, 15, 16	syl2anr 495	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( B e. A \/ A C_ B ) )
18	17	ord 392	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( -. B e. A -> A C_ B ) )
19	18	con1d 139	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A C_ B -> B e. A ) )
20		alephord 8898	. . . . 5 |- ( ( B e. On /\ A e. On ) -> ( B e. A <-> ( aleph ` B ) ~< ( aleph ` A ) ) )
21	20	ancoms 469	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( B e. A <-> ( aleph ` B ) ~< ( aleph ` A ) ) )
22		sdomnen 7984	. . . . 5 |- ( ( aleph ` B ) ~< ( aleph ` A ) -> -. ( aleph ` B ) ~~ ( aleph ` A ) )
23		sdomdom 7983	. . . . . 6 |- ( ( aleph ` B ) ~< ( aleph ` A ) -> ( aleph ` B ) ~<_ ( aleph ` A ) )
24		sbth 8080	. . . . . . 7 |- ( ( ( aleph ` B ) ~<_ ( aleph ` A ) /\ ( aleph ` A ) ~<_ ( aleph ` B ) ) -> ( aleph ` B ) ~~ ( aleph ` A ) )
25	24	ex 450	. . . . . 6 |- ( ( aleph ` B ) ~<_ ( aleph ` A ) -> ( ( aleph ` A ) ~<_ ( aleph ` B ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) )
26	23, 25	syl 17	. . . . 5 |- ( ( aleph ` B ) ~< ( aleph ` A ) -> ( ( aleph ` A ) ~<_ ( aleph ` B ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) )
27	22, 26	mtod 189	. . . 4 |- ( ( aleph ` B ) ~< ( aleph ` A ) -> -. ( aleph ` A ) ~<_ ( aleph ` B ) )
28	21, 27	syl6bi 243	. . 3 |- ( ( A e. On /\ B e. On ) -> ( B e. A -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) )
29	19, 28	syld 47	. 2 |- ( ( A e. On /\ B e. On ) -> ( -. A C_ B -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) )
30	13, 29	impcon4bid 217	1 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( aleph ` A ) ~<_ ( aleph ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Ord word 5722   Oncon0 5723   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   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-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by: (None)