MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limomss Unicode version

Theorem limomss 6705
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

Proof of Theorem limomss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limord 4942 . 2
2 ordeleqon 6624 . . 3
3 elom 6703 . . . . . . . . . 10
43simprbi 464 . . . . . . . . 9
5 limeq 4895 . . . . . . . . . . 11
6 eleq2 2530 . . . . . . . . . . 11
75, 6imbi12d 320 . . . . . . . . . 10
87spcgv 3194 . . . . . . . . 9
94, 8syl5 32 . . . . . . . 8
109com23 78 . . . . . . 7
1110imp 429 . . . . . 6
1211ssrdv 3509 . . . . 5
1312ex 434 . . . 4
14 omsson 6704 . . . . . 6
15 sseq2 3525 . . . . . 6
1614, 15mpbiri 233 . . . . 5
1716a1d 25 . . . 4
1813, 17jaoi 379 . . 3
192, 18sylbi 195 . 2
201, 19mpcom 36 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  C_wss 3475  Ordword 4882   con0 4883  Limwlim 4884   com 6700
This theorem is referenced by:  limom  6715  rdg0  7106  frfnom  7119  frsuc  7121  r1fin  8212  rankdmr1  8240  rankeq0b  8299  cardlim  8374  ackbij2  8644  cfom  8665  wunom  9119  inar1  9174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-om 6701
  Copyright terms: Public domain W3C validator