Theorem omsubel 5883

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

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 5881	. . 3 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))
2		sbth 5520	. . . . . . . . . . 11 |- (((aleph` A) ~<_ (aleph` B) /\ (aleph` B) ~<_ (aleph` A)) -> (aleph` A) ~~ (aleph` B))
3		fvex 4689	. . . . . . . . . . . 12 |- (aleph` B) e. _V
4		ssdomg 5467	. . . . . . . . . . . 12 |- ((aleph` B) e. _V -> ((aleph` B) C_ (aleph` A) -> (aleph` B) ~<_ (aleph` A)))
5	3, 4	ax-mp 7	. . . . . . . . . . 11 |- ((aleph` B) C_ (aleph` A) -> (aleph` B) ~<_ (aleph` A))
6	2, 5	sylan2 500	. . . . . . . . . 10 |- (((aleph` A) ~<_ (aleph` B) /\ (aleph` B) C_ (aleph` A)) -> (aleph` A) ~~ (aleph` B))
7	6	expcom 403	. . . . . . . . 9 |- ((aleph` B) C_ (aleph` A) -> ((aleph` A) ~<_ (aleph` B) -> (aleph` A) ~~ (aleph` B)))
8	7	adantl 424	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ (aleph` B) C_ (aleph` A)) -> ((aleph` A) ~<_ (aleph` B) -> (aleph` A) ~~ (aleph` B)))
9		sdomdom 5445	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) -> (aleph` A) ~<_ (aleph` B))
10	8, 9	syl5 20	. . . . . . 7 |- (((A e. On /\ B e. On) /\ (aleph` B) C_ (aleph` A)) -> ((aleph` A) ~< (aleph` B) -> (aleph` A) ~~ (aleph` B)))
11		sdomnen 5446	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B))
12	11	a1i 8	. . . . . . 7 |- (((A e. On /\ B e. On) /\ (aleph` B) C_ (aleph` A)) -> ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B)))
13	10, 12	pm2.65d 151	. . . . . 6 |- (((A e. On /\ B e. On) /\ (aleph` B) C_ (aleph` A)) -> -. (aleph` A) ~< (aleph` B))
14	13	ex 402	. . . . 5 |- ((A e. On /\ B e. On) -> ((aleph` B) C_ (aleph` A) -> -. (aleph` A) ~< (aleph` B)))
15		alephon 5876	. . . . . . . 8 |- (aleph` A) e. On
16		eloni 3667	. . . . . . . 8 |- ((aleph` A) e. On -> Ord (aleph` A))
17	15, 16	ax-mp 7	. . . . . . 7 |- Ord (aleph` A)
18		alephon 5876	. . . . . . . 8 |- (aleph` B) e. On
19		eloni 3667	. . . . . . . 8 |- ((aleph` B) e. On -> Ord (aleph` B))
20	18, 19	ax-mp 7	. . . . . . 7 |- Ord (aleph` B)
21		ordtri2or 3766	. . . . . . 7 |- ((Ord (aleph` A) /\ Ord (aleph` B)) -> ((aleph` A) e. (aleph` B) \/ (aleph` B) C_ (aleph` A)))
22	17, 20, 21	mp2an 761	. . . . . 6 |- ((aleph` A) e. (aleph` B) \/ (aleph` B) C_ (aleph` A))
23	22	ori 247	. . . . 5 |- (-. (aleph` A) e. (aleph` B) -> (aleph` B) C_ (aleph` A))
24	14, 23	syl5 20	. . . 4 |- ((A e. On /\ B e. On) -> (-. (aleph` A) e. (aleph` B) -> -. (aleph` A) ~< (aleph` B)))
25	24	con4d 91	. . 3 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< (aleph` B) -> (aleph` A) e. (aleph` B)))
26	1, 25	sylbid 220	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) e. (aleph` B)))
27		onelss 3705	. . . . . . . 8 |- ((aleph` B) e. On -> ((aleph` A) e. (aleph` B) -> (aleph` A) C_ (aleph` B)))
28	18, 27	ax-mp 7	. . . . . . 7 |- ((aleph` A) e. (aleph` B) -> (aleph` A) C_ (aleph` B))
29		ssdomg 5467	. . . . . . . 8 |- ((aleph` A) e. On -> ((aleph` A) C_ (aleph` B) -> (aleph` A) ~<_ (aleph` B)))
30	15, 29	ax-mp 7	. . . . . . 7 |- ((aleph` A) C_ (aleph` B) -> (aleph` A) ~<_ (aleph` B))
31	28, 30	syl 12	. . . . . 6 |- ((aleph` A) e. (aleph` B) -> (aleph` A) ~<_ (aleph` B))
32	31	adantl 424	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (aleph` A) ~<_ (aleph` B))
33		omsubdom 5882	. . . . . 6 |- ((A e. On /\ B e. On) -> (A C_ B <-> (aleph` A) ~<_ (aleph` B)))
34	33	adantr 425	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A C_ B <-> (aleph` A) ~<_ (aleph` B)))
35	32, 34	mpbird 213	. . . 4 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> A C_ B)
36		ordsseleq 3687	. . . . . . 7 |- ((Ord A /\ Ord B) -> (A C_ B <-> (A e. B \/ A = B)))
37		eloni 3667	. . . . . . 7 |- (A e. On -> Ord A)
38		eloni 3667	. . . . . . 7 |- (B e. On -> Ord B)
39	36, 37, 38	syl2an 503	. . . . . 6 |- ((A e. On /\ B e. On) -> (A C_ B <-> (A e. B \/ A = B)))
40	39	adantr 425	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A C_ B <-> (A e. B \/ A = B)))
41		idd 75	. . . . . 6 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A e. B -> A e. B))
42		ordirr 3676	. . . . . . . . . . . 12 |- (Ord (aleph` A) -> -. (aleph` A) e. (aleph` A))
43	17, 42	ax-mp 7	. . . . . . . . . . 11 |- -. (aleph` A) e. (aleph` A)
44		fveq2 4681	. . . . . . . . . . . 12 |- (A = B -> (aleph` A) = (aleph` B))
45	44	eleq2d 1964	. . . . . . . . . . 11 |- (A = B -> ((aleph` A) e. (aleph` A) <-> (aleph` A) e. (aleph` B)))
46	43, 45	mtbii 784	. . . . . . . . . 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 107	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((aleph` A) e. (aleph` B) -> -. A = B))
49	48	imp 377	. . . . . . 7 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> -. A = B)
50	49	pm2.21d 94	. . . . . 6 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A = B -> A e. B))
51	41, 50	jaod 469	. . . . 5 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> ((A e. B \/ A = B) -> A e. B))
52	40, 51	sylbid 220	. . . 4 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> (A C_ B -> A e. B))
53	35, 52	mpd 29	. . 3 |- (((A e. On /\ B e. On) /\ (aleph` A) e. (aleph` B)) -> A e. B)
54	53	ex 402	. 2 |- ((A e. On /\ B e. On) -> ((aleph` A) e. (aleph` B) -> A e. B))
55	26, 54	impbid 574	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) e. (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:  omsubss 5884  omsubindss 5888  omsubssOLD 15393  omsubindssOLD 15397

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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