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Theorem epweon 7486
Description: The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. 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
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onfr 6223 . 2 E Fr On
2 eloni 6194 . . . 4 (𝑥 ∈ On → Ord 𝑥)
3 eloni 6194 . . . 4 (𝑦 ∈ On → Ord 𝑦)
4 ordtri3or 6216 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
5 epel 5462 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 262 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 5462 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1148 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
94, 8sylibr 235 . . . 4 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
102, 3, 9syl2an 595 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1110rgen2 3200 . 2 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)
12 dfwe2 7485 . 2 ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
131, 11, 12mpbir2an 707 1 E We On
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3o 1078  wcel 2105  wral 3135   class class class wbr 5057   E cep 5457   Fr wfr 5504   We wwe 5506  Ord word 6183  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  ordon  7487  omsinds  7589  onnseq  7970  dfrecs3  7998  tfr1ALT  8025  tfr2ALT  8026  tfr3ALT  8027  ordunifi  8756  ordtypelem8  8977  oismo  8992  cantnfcl  9118  leweon  9425  r0weon  9426  ac10ct  9448  dfac12lem2  9558  cflim2  9673  cofsmo  9679  hsmexlem1  9836  smobeth  9996  gruina  10228  ltsopi  10298  dford5  32854  finminlem  33563  dnwech  39526  aomclem4  39535
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