Theorem omsubel 11444

Description: Relationship between ordering of ordinal numbers and ordering of infinite initial ordinals.

Assertion

Ref	Expression
omsubel	|- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) e. (aleph` B)))

Proof of Theorem omsubel

Step	Hyp	Ref	Expression
1		omsubsdom 11442	. . 3 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))
2		sbth 4602	. . . . . . . . . . 11 |- (((aleph` A) ~<_ (aleph` B) /\ (aleph` B) ~<_ (aleph` A)) -> (aleph` A) ~~ (aleph` B))
3		fvex 3843	. . . . . . . . . . . 12 |- (aleph` B) e. V
4		ssdomg 4549	. . . . . . . . . . . 12 |- ((aleph` B) e. V -> ((aleph` B) (_ (aleph` A) -> (aleph` B) ~<_ (aleph` A)))
5	3, 4	ax-mp 7	. . . . . . . . . . 11 |- ((aleph` B) (_ (aleph` A) -> (aleph` B) ~<_ (aleph` A))
6	2, 5	sylan2 453	. . . . . . . . . 10 |- (((aleph` A) ~<_ (aleph` B) /\ (aleph` B) (_ (aleph` A)) -> (aleph` A) ~~ (aleph` B))
7	6	expcom 372	. . . . . . . . 9 |- ((aleph` B) (_ (aleph` A) -> ((aleph` A) ~<_ (aleph` B) -> (aleph` A) ~~ (aleph` B)))
8	7	adantl 388	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ (aleph` B) (_ (aleph` A)) -> ((aleph` A) ~<_ (aleph` B) -> (aleph` A) ~~ (aleph` B)))
9		sdomdom 4527	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) -> (aleph` A) ~<_ (aleph` B))
10	8, 9	syl5 21	. . . . . . 7 |- (((A e. On /\ B e. On) /\ (aleph` B) (_ (aleph` A)) -> ((aleph` A) ~< (aleph` B) -> (aleph` A) ~~ (aleph` B)))
11		sdomnen 4528	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B))
12	11	a1i 8	. . . . . . 7 |- (((A e. On /\ B e. On) /\ (aleph` B) (_ (aleph` A)) -> ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B)))
13	10, 12	pm2.65d 134	. . . . . 6 |- (((A e. On /\ B e. On) /\ (aleph` B) (_ (aleph` A)) -> -. (aleph` A) ~< (aleph` B))
14	13	ex 371	. . . . 5 |- ((A e. On /\ B e. On) -> ((aleph` B) (_ (aleph` A) -> -. (aleph` A) ~< (aleph` B)))
15		alephon 5015	. . . . . . . 8 |- (aleph` A) e. On
16		eloni 2985	. . . . . . . 8 |- ((aleph` A) e. On -> Ord (aleph` A))
17	15, 16	ax-mp 7	. . . . . . 7 |- Ord (aleph` A)
18		alephon 5015	. . . . . . . 8 |- (aleph` B) e. On
19		eloni 2985	. . . . . . . 8 |- ((aleph` B) e. On -> Ord (aleph` B))
20	18, 19	ax-mp 7	. . . . . . 7 |- Ord (aleph` B)
21		ordtri2or 3067	. . . . . . 7 |- ((Ord (aleph` A) /\ Ord (aleph` B)) -> ((aleph` A) e. (aleph` B) \/ (aleph` B) (_ (aleph` A)))
22	17, 20, 21	mp2an 701	. . . . . 6 |- ((aleph` A) e. (aleph` B) \/ (aleph` B) (_ (aleph` A))
23	22	ori 228	. . . . 5 |- (-. (aleph` A) e. (aleph` B) -> (aleph` B) (_ (aleph` A))
24	14, 23	syl5 21	. . . 4 |- ((A e. On /\ B e. On) -> (-. (aleph` A) e. (aleph` B) -> -. (aleph` A) ~< (aleph` B)))
25	24	con4d 75	. . 3 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< (aleph` B) -> (aleph` A) e. (aleph` B)))
26	1, 25	sylbid 201	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) e. (aleph` B)))
27		onelss 3017	. . . . . . . 8 |- ((aleph` B) e. On -> ((aleph` A) e. (aleph` B) -> (aleph` A) (_ (aleph` B)))
28	18, 27	ax-mp 7	. . . . . . 7 |- ((aleph` A) e. (aleph` B) -> (aleph` A) (_ (aleph` B))
29		ssdomg 4549	. . . . . . . 8 |- ((aleph` A) e. On -> ((aleph` A) (_ (aleph` B) -> (aleph` A) ~<_ (aleph` B)))
30	15, 29	ax-mp 7	. . . . . . 7 |- ((aleph` A) (_ (aleph` B) -> (aleph` A) ~<_ (aleph` B))
31	28, 30	syl 10	. . . . . 6 |- ((aleph` A) e. (aleph` B) -> (aleph` A) ~<_ (aleph` B))
32	31	adantl 388	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (aleph` A) ~<_ (aleph` B))
33		omsubdom 11443	. . . . . 6 |- ((A e. On /\ B e. On) -> (A (_ B <-> (aleph` A) ~<_ (aleph` B)))
34	33	adantr 389	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A (_ B <-> (aleph` A) ~<_ (aleph` B)))
35	32, 34	mpbird 194	. . . 4 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> A (_ B)
36		ordsseleq 3004	. . . . . . 7 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))
37		eloni 2985	. . . . . . 7 |- (A e. On -> Ord A)
38		eloni 2985	. . . . . . 7 |- (B e. On -> Ord B)
39	36, 37, 38	syl2an 456	. . . . . 6 |- ((A e. On /\ B e. On) -> (A (_ B <-> (A e. B \/ A = B)))
40	39	adantr 389	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A (_ B <-> (A e. B \/ A = B)))
41		idd 61	. . . . . 6 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A e. B -> A e. B))
42		ordirr 2993	. . . . . . . . . . . 12 |- (Ord (aleph` A) -> -. (aleph` A) e. (aleph` A))
43	17, 42	ax-mp 7	. . . . . . . . . . 11 |- -. (aleph` A) e. (aleph` A)
44		fveq2 3835	. . . . . . . . . . . 12 |- (A = B -> (aleph` A) = (aleph` B))
45	44	eleq2d 1584	. . . . . . . . . . 11 |- (A = B -> ((aleph` A) e. (aleph` A) <-> (aleph` A) e. (aleph` B)))
46	43, 45	mtbii 721	. . . . . . . . . 10 |- (A = B -> -. (aleph` A) e. (aleph` B))
47	46	a1i 8	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (A = B -> -. (aleph` A) e. (aleph` B)))
48	47	con2d 91	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((aleph` A) e. (aleph` B) -> -. A = B))
49	48	imp 348	. . . . . . 7 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> -. A = B)
50	49	pm2.21d 78	. . . . . 6 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A = B -> A e. B))
51	41, 50	jaod 424	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> ((A e. B \/ A = B) -> A e. B))
52	40, 51	sylbid 201	. . . 4 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A (_ B -> A e. B))
53	35, 52	mpd 26	. . 3 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> A e. B)
54	53	ex 371	. 2 |- ((A e. On /\ B e. On) -> ((aleph` A) e. (aleph` B) -> A e. B))
55	26, 54	impbid 519	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) e. (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857   (_ wss 2099   class class class wbr 2692  Ord word 2974  Oncon0 2975  ` cfv 3263   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507  alephcale 4960

This theorem is referenced by:  omsubss 11445  omsubindss 11449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-iso 3280  df-oprab 4024  df-rdg 4233  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-aleph 4963

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