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Theorem dford5reg 35764
Description: Given ax-reg 9630, an ordinal is a transitive class totally ordered by the membership relation. (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 6389 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 9643 . . . 4 E Fr 𝐴
3 df-we 5643 . . . 4 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
42, 3mpbiran 709 . . 3 ( E We 𝐴 ↔ E Or 𝐴)
54anbi2i 623 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
61, 5bitri 275 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  Tr wtr 5265   E cep 5588   Or wor 5596   Fr wfr 5638   We wwe 5640  Ord word 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641  df-we 5643  df-ord 6389
This theorem is referenced by: (None)
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