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Theorem dford5reg 30724
Description: Given ax-reg 8357, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
Assertion
Ref Expression
dford5reg (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Proof of Theorem dford5reg
StepHypRef Expression
1 df-ord 5628 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 8369 . . . 4 E Fr 𝐴
3 df-we 4988 . . . 4 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
42, 3mpbiran 954 . . 3 ( E We 𝐴 ↔ E Or 𝐴)
54anbi2i 725 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
61, 5bitri 262 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  Tr wtr 4674   E cep 4936   Or wor 4947   Fr wfr 4983   We wwe 4985  Ord word 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827  ax-reg 8357
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-eprel 4938  df-fr 4986  df-we 4988  df-ord 5628
This theorem is referenced by: (None)
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