Theorem limomss 3956

Description: The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity.

Assertion

Ref	Expression
limomss	|- (Lim A -> om C_ A)

Proof of Theorem limomss

Step	Hyp	Ref	Expression
1		limord 3723	. 2 |- (Lim A -> Ord A)
2		ordeleqon 3866	. . 3 |- (Ord A <-> (A e. On \/ A = On))
3		limeq 3669	. . . . . . . . . . 11 |- (y = A -> (Lim y <-> Lim A))
4		eleq2 1958	. . . . . . . . . . 11 |- (y = A -> (x e. y <-> x e. A))
5	3, 4	imbi12d 688	. . . . . . . . . 10 |- (y = A -> ((Lim y -> x e. y) <-> (Lim A -> x e. A)))
6	5	cla4gv 2364	. . . . . . . . 9 |- (A e. On -> (A.y(Lim y -> x e. y) -> (Lim A -> x e. A)))
7		visset 2295	. . . . . . . . . . 11 |- x e. _V
8	7	elom 3952	. . . . . . . . . 10 |- (x e. om <-> (Ord x /\ A.y(Lim y -> x e. y)))
9	8	simprbi 353	. . . . . . . . 9 |- (x e. om -> A.y(Lim y -> x e. y))
10	6, 9	syl5 20	. . . . . . . 8 |- (A e. On -> (x e. om -> (Lim A -> x e. A)))
11	10	com23 36	. . . . . . 7 |- (A e. On -> (Lim A -> (x e. om -> x e. A)))
12	11	imp 377	. . . . . 6 |- ((A e. On /\ Lim A) -> (x e. om -> x e. A))
13	12	ssrdv 2622	. . . . 5 |- ((A e. On /\ Lim A) -> om C_ A)
14	13	ex 402	. . . 4 |- (A e. On -> (Lim A -> om C_ A))
15		omsson 3954	. . . . . 6 |- om C_ On
16		sseq2 2639	. . . . . 6 |- (A = On -> (om C_ A <-> om C_ On))
17	15, 16	mpbiri 211	. . . . 5 |- (A = On -> om C_ A)
18	17	a1d 15	. . . 4 |- (A = On -> (Lim A -> om C_ A))
19	14, 18	jaoi 368	. . 3 |- ((A e. On \/ A = On) -> (Lim A -> om C_ A))
20	2, 19	sylbi 216	. 2 |- (Ord A -> (Lim A -> om C_ A))
21	1, 20	mpcom 60	1 |- (Lim A -> om C_ A)

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300   C_ wss 2593  Ord word 3656  Oncon0 3657  Lim wlim 3658  omcom 3949

This theorem is referenced by:  cardlim 6003

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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  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

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