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

Theorem epweon 6557
Description: The epsilon relation well-orders the class of ordinal numbers. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.)
Assertion
Ref Expression
epweon  |-  _E  We  On

Proof of Theorem epweon
StepHypRef Expression
1 ordon 6556 . 2  |-  Ord  On
2 ordwe 4834 . 2  |-  ( Ord 
On  ->  _E  We  On )
31, 2ax-mp 5 1  |-  _E  We  On
Colors of variables: wff setvar class
Syntax hints:    _E cep 4731    We wwe 4780   Ord word 4820   Oncon0 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825
This theorem is referenced by:  onnseq  6972  ordunifi  7724  ordtypelem8  7904  oismo  7919  cantnfcl  8038  cantnfclOLD  8068  leweon  8341  r0weon  8342  ac10ct  8367  dfac12lem2  8476  cflim2  8595  cofsmo  8601  hsmexlem1  8758  smobeth  8913  gruina  9146  ltsopi  9216  omsinds  29972  tfrALTlem  30035  tfr1ALT  30036  tfr2ALT  30037  tfr3ALT  30038  finminlem  30534  dnwech  35337  aomclem4  35346
  Copyright terms: Public domain W3C validator