Theorem limomss 2378

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 (_ A)

Proof of Theorem limomss

Step	Hyp	Ref	Expression
1		limord 2283	. 2 |- (Lim A -> Ord A)
2		ordeleqon 2241	. . 3 |- (Ord A <-> (A e. On \/ A = On))
3		limeq 2211	. . . . . . . . . . 11 |- (y = A -> (Lim y <-> Lim A))
4		eleq2 1150	. . . . . . . . . . 11 |- (y = A -> (x e. y <-> x e. A))
5	3, 4	imbi12d 474	. . . . . . . . . 10 |- (y = A -> ((Lim y -> x e. y) <-> (Lim A -> x e. A)))
6	5	cla4gv 1396	. . . . . . . . 9 |- (A e. On -> (A.y(Lim y -> x e. y) -> (Lim A -> x e. A)))
7		visset 1350	. . . . . . . . . . 11 |- x e. V
8	7	elom 2375	. . . . . . . . . 10 |- (x e. om <-> (Ord x /\ A.y(Lim y -> x e. y)))
9	8	pm3.27bd 263	. . . . . . . . 9 |- (x e. om -> A.y(Lim y -> x e. y))
10	6, 9	syl5 22	. . . . . . . 8 |- (A e. On -> (x e. om -> (Lim A -> x e. A)))
11	10	com23 32	. . . . . . 7 |- (A e. On -> (Lim A -> (x e. om -> x e. A)))
12	11	imp 277	. . . . . 6 |- ((A e. On /\ Lim A) -> (x e. om -> x e. A))
13	12	ssrdv 1509	. . . . 5 |- ((A e. On /\ Lim A) -> om (_ A)
14	13	exp 291	. . . 4 |- (A e. On -> (Lim A -> om (_ A))
15		omsson 2377	. . . . . 6 |- om (_ On
16		sseq2 1522	. . . . . 6 |- (A = On -> (om (_ A <-> om (_ On))
17	15, 16	mpbiri 169	. . . . 5 |- (A = On -> om (_ A)
18	17	a1d 14	. . . 4 |- (A = On -> (Lim A -> om (_ A))
19	14, 18	jaoi 275	. . 3 |- ((A e. On \/ A = On) -> (Lim A -> om (_ A))
20	2, 19	sylbi 174	. 2 |- (Ord A -> (Lim A -> om (_ A))
21	1, 20	mpcom 49	1 |- (Lim A -> om (_ A)

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196  A.wal 672   e. wel 803   = wceq 1091   e. wcel 1092   (_ wss 1487  Ord word 2198  Oncon0 2199  Lim wlim 2200  omcom 2372

This theorem is referenced by:  cardlim 3657

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-om 2373

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