Theorem limomss 3143

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 3034	. 2 |- (Lim A -> Ord A)
2		ordeleqon 2996	. . 3 |- (Ord A <-> (A e. On \/ A = On))
3		limeq 2966	. . . . . . . . . . 11 |- (y = A -> (Lim y <-> Lim A))
4		eleq2 1538	. . . . . . . . . . 11 |- (y = A -> (x e. y <-> x e. A))
5	3, 4	imbi12d 628	. . . . . . . . . 10 |- (y = A -> ((Lim y -> x e. y) <-> (Lim A -> x e. A)))
6	5	cla4gv 1865	. . . . . . . . 9 |- (A e. On -> (A.y(Lim y -> x e. y) -> (Lim A -> x e. A)))
7		visset 1816	. . . . . . . . . . 11 |- x e. V
8	7	elom 3140	. . . . . . . . . 10 |- (x e. om <-> (Ord x /\ A.y(Lim y -> x e. y)))
9	8	pm3.27bi 326	. . . . . . . . 9 |- (x e. om -> A.y(Lim y -> x e. y))
10	6, 9	syl5 21	. . . . . . . 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 350	. . . . . 6 |- ((A e. On /\ Lim A) -> (x e. om -> x e. A))
13	12	ssrdv 2073	. . . . 5 |- ((A e. On /\ Lim A) -> om (_ A)
14	13	ex 373	. . . 4 |- (A e. On -> (Lim A -> om (_ A))
15		omsson 3142	. . . . . 6 |- om (_ On
16		sseq2 2086	. . . . . 6 |- (A = On -> (om (_ A <-> om (_ On))
17	15, 16	mpbiri 194	. . . . 5 |- (A = On -> om (_ A)
18	17	a1d 12	. . . 4 |- (A = On -> (Lim A -> om (_ A))
19	14, 18	jaoi 341	. . 3 |- ((A e. On \/ A = On) -> (Lim A -> om (_ A))
20	2, 19	sylbi 199	. 2 |- (Ord A -> (Lim A -> om (_ A))
21	1, 20	mpcom 49	1 |- (Lim A -> om (_ A)

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960   (_ wss 2050  Ord word 2953  Oncon0 2954  Lim wlim 2955  omcom 3137

This theorem is referenced by:  cardlim 4862

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-om 3138

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