Theorem omsubsdom 11442

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 5015, which does not use Choice.)

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 11441	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))
2		ordtri2 3010	. . . . . 6 |- ((Ord A /\ Ord B) -> (A e. B <-> -. (A = B \/ B e. A)))
3		eloni 2985	. . . . . 6 |- (A e. On -> Ord A)
4		eloni 2985	. . . . . 6 |- (B e. On -> Ord B)
5	2, 3, 4	syl2an 456	. . . . 5 |- ((A e. On /\ B e. On) -> (A e. B <-> -. (A = B \/ B e. A)))
6	5	con2bid 529	. . . 4 |- ((A e. On /\ B e. On) -> ((A = B \/ B e. A) <-> -. A e. B))
7	6	bicomd 524	. . 3 |- ((A e. On /\ B e. On) -> (-. A e. B <-> (A = B \/ B e. A)))
8		eqeng 4533	. . . . . . 7 |- ((aleph` A) e. V -> ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B)))
9		fvex 3843	. . . . . . . 8 |- (aleph` A) e. V
10	9	a1i 8	. . . . . . 7 |- (((A e. On /\ B e. On) /\ A = B) -> (aleph` A) e. V)
11		fveq2 3835	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
12	11	adantl 388	. . . . . . 7 |- (((A e. On /\ B e. On) /\ A = B) -> (aleph` A) = (aleph` B))
13	8, 10, 12	sylc 68	. . . . . 6 |- (((A e. On /\ B e. On) /\ A = B) -> (aleph` A) ~~ (aleph` B))
14		brsdom 4522	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) <-> ((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)))
15	14	pm3.27bi 324	. . . . . . 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 110	. . . . 5 |- (((A e. On /\ B e. On) /\ A = B) -> -. (aleph` A) ~< (aleph` B))
18	17	ex 371	. . . 4 |- ((A e. On /\ B e. On) -> (A = B -> -. (aleph` A) ~< (aleph` B)))
19		sbth 4602	. . . . . . . 8 |- (((aleph` A) ~<_ (aleph` B) /\ (aleph` B) ~<_ (aleph` A)) -> (aleph` A) ~~ (aleph` B))
20		sdomdom 4527	. . . . . . . 8 |- ((aleph` A) ~< (aleph` B) -> (aleph` A) ~<_ (aleph` B))
21		omsubsdomlem2 11441	. . . . . . . . . . 11 |- ((B e. On /\ A e. On) -> (B e. A -> (aleph` B) ~< (aleph` A)))
22		sdomdom 4527	. . . . . . . . . . 11 |- ((aleph` B) ~< (aleph` A) -> (aleph` B) ~<_ (aleph` A))
23	21, 22	syl6 22	. . . . . . . . . 10 |- ((B e. On /\ A e. On) -> (B e. A -> (aleph` B) ~<_ (aleph` A)))
24	23	ancoms 438	. . . . . . . . 9 |- ((A e. On /\ B e. On) -> (B e. A -> (aleph` B) ~<_ (aleph` A)))
25	24	imp 348	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ B e. A) -> (aleph` B) ~<_ (aleph` A))
26	19, 20, 25	syl2an 456	. . . . . . 7 |- (((aleph` A) ~< (aleph` B) /\ ((A e. On /\ B e. On) /\ B e. A)) -> (aleph` A) ~~ (aleph` B))
27	26	expcom 372	. . . . . 6 |- (((A e. On /\ B e. On) /\ B e. A) -> ((aleph` A) ~< (aleph` B) -> (aleph` A) ~~ (aleph` B)))
28		sdomnen 4528	. . . . . . 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 134	. . . . 5 |- (((A e. On /\ B e. On) /\ B e. A) -> -. (aleph` A) ~< (aleph` B))
31	30	ex 371	. . . 4 |- ((A e. On /\ B e. On) -> (B e. A -> -. (aleph` A) ~< (aleph` B)))
32	18, 31	jaod 424	. . 3 |- ((A e. On /\ B e. On) -> ((A = B \/ B e. A) -> -. (aleph` A) ~< (aleph` B)))
33	7, 32	sylbid 201	. 2 |- ((A e. On /\ B e. On) -> (-. A e. B -> -. (aleph` A) ~< (aleph` B)))
34	1, 33	impbid3 11338	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857   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:  omsubdom 11443  omsubel 11444

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

Copyright terms: Public domain