Theorem epweon 7025

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

Step	Hyp	Ref	Expression
1		ordon 7024	. 2 ⊢ Ord On
2		ordwe 5774	. 2 ⊢ (Ord On → E We On)
3	1, 2	ax-mp 5	1 ⊢ E We On

Syntax hints:   E cep 5057   We wwe 5101  Ord word 5760  Oncon0 5761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765

This theorem is referenced by:  omsinds  7126  onnseq  7486  dfrecs3  7514  tfr1ALT  7541  tfr2ALT  7542  tfr3ALT  7543  ordunifi  8251  ordtypelem8  8471  oismo  8486  cantnfcl  8602  leweon  8872  r0weon  8873  ac10ct  8895  dfac12lem2  9004  cflim2  9123  cofsmo  9129  hsmexlem1  9286  smobeth  9446  gruina  9678  ltsopi  9748  dford5  31734  finminlem  32437  dnwech  37935  aomclem4  37944