Theorem omsubsdom 5881

Description: Relationship between ordering on ordinal numbers and strict dominance of infinite initial ordinal numbers, which are frequently denoted by omega with an ordinal number subscript. The aleph notation is being recycled for this purpose, although the aleph function does not have the same meaning if Choice is not assumed. (Closure is given by alephon 5876, which does not use Choice.) (Contributed by Jeff Hankins, 23-Oct-2009.)

Assertion

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

Proof of Theorem omsubsdom

Step	Hyp	Ref	Expression
1		omsubsdomlem2 5880	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))
2		ordtri2 3696	. . . . . 6 |- ((Ord A /\ Ord B) -> (A e. B <-> -. (A = B \/ B e. A)))
3		eloni 3667	. . . . . 6 |- (A e. On -> Ord A)
4		eloni 3667	. . . . . 6 |- (B e. On -> Ord B)
5	2, 3, 4	syl2an 503	. . . . 5 |- ((A e. On /\ B e. On) -> (A e. B <-> -. (A = B \/ B e. A)))
6	5	con2bid 585	. . . 4 |- ((A e. On /\ B e. On) -> ((A = B \/ B e. A) <-> -. A e. B))
7	6	bicomd 580	. . 3 |- ((A e. On /\ B e. On) -> (-. A e. B <-> (A = B \/ B e. A)))
8		fvex 4689	. . . . . . . 8 |- (aleph` A) e. _V
9	8	a1i 8	. . . . . . 7 |- (((A e. On /\ B e. On) /\ A = B) -> (aleph` A) e. _V)
10		fveq2 4681	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
11	10	adantl 424	. . . . . . 7 |- (((A e. On /\ B e. On) /\ A = B) -> (aleph` A) = (aleph` B))
12		eqeng 5451	. . . . . . 7 |- ((aleph` A) e. _V -> ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B)))
13	9, 11, 12	sylc 83	. . . . . 6 |- (((A e. On /\ B e. On) /\ A = B) -> (aleph` A) ~~ (aleph` B))
14		brsdom 5440	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) <-> ((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)))
15	14	simprbi 353	. . . . . . 7 |- ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B))
16	15	a1i 8	. . . . . 6 |- (((A e. On /\ B e. On) /\ A = B) -> ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B)))
17	13, 16	mt2d 126	. . . . 5 |- (((A e. On /\ B e. On) /\ A = B) -> -. (aleph` A) ~< (aleph` B))
18	17	ex 402	. . . 4 |- ((A e. On /\ B e. On) -> (A = B -> -. (aleph` A) ~< (aleph` B)))
19		sbth 5520	. . . . . . . 8 |- (((aleph` A) ~<_ (aleph` B) /\ (aleph` B) ~<_ (aleph` A)) -> (aleph` A) ~~ (aleph` B))
20		sdomdom 5445	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) -> (aleph` A) ~<_ (aleph` B))
21		omsubsdomlem2 5880	. . . . . . . . . . 11 |- ((B e. On /\ A e. On) -> (B e. A -> (aleph` B) ~< (aleph` A)))
22		sdomdom 5445	. . . . . . . . . . 11 |- ((aleph` B) ~< (aleph` A) -> (aleph` B) ~<_ (aleph` A))
23	21, 22	syl6 25	. . . . . . . . . 10 |- ((B e. On /\ A e. On) -> (B e. A -> (aleph` B) ~<_ (aleph` A)))
24	23	ancoms 484	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (B e. A -> (aleph` B) ~<_ (aleph` A)))
25	24	imp 377	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ B e. A) -> (aleph` B) ~<_ (aleph` A))
26	19, 20, 25	syl2an 503	. . . . . . 7 |- (((aleph` A) ~< (aleph` B) /\ ((A e. On /\ B e. On) /\ B e. A)) -> (aleph` A) ~~ (aleph` B))
27	26	expcom 403	. . . . . 6 |- (((A e. On /\ B e. On) /\ B e. A) -> ((aleph` A) ~< (aleph` B) -> (aleph` A) ~~ (aleph` B)))
28		sdomnen 5446	. . . . . . 7 |- ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B))
29	28	a1i 8	. . . . . 6 |- (((A e. On /\ B e. On) /\ B e. A) -> ((aleph` A) ~< (aleph` B) -> -. (aleph` A) ~~ (aleph` B)))
30	27, 29	pm2.65d 151	. . . . 5 |- (((A e. On /\ B e. On) /\ B e. A) -> -. (aleph` A) ~< (aleph` B))
31	30	ex 402	. . . 4 |- ((A e. On /\ B e. On) -> (B e. A -> -. (aleph` A) ~< (aleph` B)))
32	18, 31	jaod 469	. . 3 |- ((A e. On /\ B e. On) -> ((A = B \/ B e. A) -> -. (aleph` A) ~< (aleph` B)))
33	7, 32	sylbid 220	. 2 |- ((A e. On /\ B e. On) -> (-. A e. B -> -. (aleph` A) ~< (aleph` B)))
34	1, 33	impcon4bid 578	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   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:  omsubdom 5882  omsubel 5883  omsubdomOLD 15391  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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