Theorem omsubdom 5882

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

Assertion

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

Proof of Theorem omsubdom

Step	Hyp	Ref	Expression
1		ordsseleq 3687	. . . 4 |- ((Ord A /\ Ord B) -> (A C_ B <-> (A e. B \/ A = B)))
2		eloni 3667	. . . 4 |- (A e. On -> Ord A)
3		eloni 3667	. . . 4 |- (B e. On -> Ord B)
4	1, 2, 3	syl2an 503	. . 3 |- ((A e. On /\ B e. On) -> (A C_ B <-> (A e. B \/ A = B)))
5		omsubsdom 5881	. . . . 5 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))
6		sdomdom 5445	. . . . 5 |- ((aleph` A) ~< (aleph` B) -> (aleph` A) ~<_ (aleph` B))
7	5, 6	syl6bi 231	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~<_ (aleph` B)))
8		fveq2 4681	. . . . . . 7 |- (A = B -> (aleph` A) = (aleph` B))
9		fvex 4689	. . . . . . . 8 |- (aleph` A) e. _V
10		eqeng 5451	. . . . . . . 8 |- ((aleph` A) e. _V -> ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B)))
11	9, 10	ax-mp 7	. . . . . . 7 |- ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B))
12	8, 11	syl 12	. . . . . 6 |- (A = B -> (aleph` A) ~~ (aleph` B))
13	12	a1i 8	. . . . 5 |- ((A e. On /\ B e. On) -> (A = B -> (aleph` A) ~~ (aleph` B)))
14		endom 5444	. . . . 5 |- ((aleph` A) ~~ (aleph` B) -> (aleph` A) ~<_ (aleph` B))
15	13, 14	syl6 25	. . . 4 |- ((A e. On /\ B e. On) -> (A = B -> (aleph` A) ~<_ (aleph` B)))
16	7, 15	jaod 469	. . 3 |- ((A e. On /\ B e. On) -> ((A e. B \/ A = B) -> (aleph` A) ~<_ (aleph` B)))
17	4, 16	sylbid 220	. 2 |- ((A e. On /\ B e. On) -> (A C_ B -> (aleph` A) ~<_ (aleph` B)))
18		ordtri2or 3766	. . . . . . 7 |- ((Ord B /\ Ord A) -> (B e. A \/ A C_ B))
19	18, 3, 2	syl2an 503	. . . . . 6 |- ((B e. On /\ A e. On) -> (B e. A \/ A C_ B))
20	19	ancoms 484	. . . . 5 |- ((A e. On /\ B e. On) -> (B e. A \/ A C_ B))
21	20	ord 249	. . . 4 |- ((A e. On /\ B e. On) -> (-. B e. A -> A C_ B))
22	21	con1d 109	. . 3 |- ((A e. On /\ B e. On) -> (-. A C_ B -> B e. A))
23		omsubsdom 5881	. . . . 5 |- ((B e. On /\ A e. On) -> (B e. A <-> (aleph` B) ~< (aleph` A)))
24	23	ancoms 484	. . . 4 |- ((A e. On /\ B e. On) -> (B e. A <-> (aleph` B) ~< (aleph` A)))
25		sdomnen 5446	. . . . 5 |- ((aleph` B) ~< (aleph` A) -> -. (aleph` B) ~~ (aleph` A))
26		sdomdom 5445	. . . . . 6 |- ((aleph` B) ~< (aleph` A) -> (aleph` B) ~<_ (aleph` A))
27		sbth 5520	. . . . . . 7 |- (((aleph` B) ~<_ (aleph` A) /\ (aleph` A) ~<_ (aleph` B)) -> (aleph` B) ~~ (aleph` A))
28	27	ex 402	. . . . . 6 |- ((aleph` B) ~<_ (aleph` A) -> ((aleph` A) ~<_ (aleph` B) -> (aleph` B) ~~ (aleph` A)))
29	26, 28	syl 12	. . . . 5 |- ((aleph` B) ~< (aleph` A) -> ((aleph` A) ~<_ (aleph` B) -> (aleph` B) ~~ (aleph` A)))
30	25, 29	mtod 123	. . . 4 |- ((aleph` B) ~< (aleph` A) -> -. (aleph` A) ~<_ (aleph` B))
31	24, 30	syl6bi 231	. . 3 |- ((A e. On /\ B e. On) -> (B e. A -> -. (aleph` A) ~<_ (aleph` B)))
32	22, 31	syld 30	. 2 |- ((A e. On /\ B e. On) -> (-. A C_ B -> -. (aleph` A) ~<_ (aleph` B)))
33	17, 32	impcon4bid 578	1 |- ((A e. On /\ B e. On) -> (A C_ B <-> (aleph` A) ~<_ (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  Ord word 3656  Oncon0 3657  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  omsubel 5883  omsubelOLD 15392

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-oprab 4887  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-aleph 5863

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