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

Theorem epweon 6590
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 6589 . 2  |-  Ord  On
2 ordwe 4884 . 2  |-  ( Ord 
On  ->  _E  We  On )
31, 2ax-mp 5 1  |-  _E  We  On
Colors of variables: wff setvar class
Syntax hints:    _E cep 4782    We wwe 4830   Ord word 4870   Oncon0 4871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875
This theorem is referenced by:  onnseq  7005  ordunifi  7759  ordtypelem8  7939  oismo  7954  cantnfcl  8075  cantnfclOLD  8105  leweon  8378  r0weon  8379  ac10ct  8404  dfac12lem2  8513  cflim2  8632  cofsmo  8638  hsmexlem1  8795  smobeth  8950  gruina  9185  ltsopi  9255  omsinds  28862  tfrALTlem  28925  tfr1ALT  28926  tfr2ALT  28927  tfr3ALT  28928  finminlem  29700  dnwech  30587  aomclem4  30596
  Copyright terms: Public domain W3C validator