Theorem limomss 7070

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. (Contributed by NM, 30-Oct-2003.)

Assertion

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

Proof of Theorem limomss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limord 5784	. 2 |- ( Lim A -> Ord A )
2		ordeleqon 6988	. . 3 |- ( Ord A <-> ( A e. On \/ A = On ) )
3		elom 7068	. . . . . . . . . 10 |- ( x e. om <-> ( x e. On /\ A. y ( Lim y -> x e. y ) ) )
4	3	simprbi 480	. . . . . . . . 9 |- ( x e. om -> A. y ( Lim y -> x e. y ) )
5		limeq 5735	. . . . . . . . . . 11 |- ( y = A -> ( Lim y <-> Lim A ) )
6		eleq2 2690	. . . . . . . . . . 11 |- ( y = A -> ( x e. y <-> x e. A ) )
7	5, 6	imbi12d 334	. . . . . . . . . 10 |- ( y = A -> ( ( Lim y -> x e. y ) <-> ( Lim A -> x e. A ) ) )
8	7	spcgv 3293	. . . . . . . . 9 |- ( A e. On -> ( A. y ( Lim y -> x e. y ) -> ( Lim A -> x e. A ) ) )
9	4, 8	syl5 34	. . . . . . . 8 |- ( A e. On -> ( x e. om -> ( Lim A -> x e. A ) ) )
10	9	com23 86	. . . . . . 7 |- ( A e. On -> ( Lim A -> ( x e. om -> x e. A ) ) )
11	10	imp 445	. . . . . 6 |- ( ( A e. On /\ Lim A ) -> ( x e. om -> x e. A ) )
12	11	ssrdv 3609	. . . . 5 |- ( ( A e. On /\ Lim A ) -> om C_ A )
13	12	ex 450	. . . 4 |- ( A e. On -> ( Lim A -> om C_ A ) )
14		omsson 7069	. . . . . 6 |- om C_ On
15		sseq2 3627	. . . . . 6 |- ( A = On -> ( om C_ A <-> om C_ On ) )
16	14, 15	mpbiri 248	. . . . 5 |- ( A = On -> om C_ A )
17	16	a1d 25	. . . 4 |- ( A = On -> ( Lim A -> om C_ A ) )
18	13, 17	jaoi 394	. . 3 |- ( ( A e. On \/ A = On ) -> ( Lim A -> om C_ A ) )
19	2, 18	sylbi 207	. 2 |- ( Ord A -> ( Lim A -> om C_ A ) )
20	1, 19	mpcom 38	1 |- ( Lim A -> om C_ A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   C_ wss 3574   Ord word 5722   Oncon0 5723   Lim wlim 5724   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  limom  7080  rdg0  7517  frfnom  7530  frsuc  7532  r1fin  8636  rankdmr1  8664  rankeq0b  8723  cardlim  8798  ackbij2  9065  cfom  9086  wunom  9542  inar1  9597