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Theorem dford5reg 33034
 Description: Given ax-reg 9032, 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 6167 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 9044 . . . 4 E Fr 𝐴
3 df-we 5489 . . . 4 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
42, 3mpbiran 708 . . 3 ( E We 𝐴 ↔ E Or 𝐴)
54anbi2i 625 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
61, 5bitri 278 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  Tr wtr 5145   E cep 5437   Or wor 5446   Fr wfr 5484   We wwe 5486  Ord word 6163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-reg 9032 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-eprel 5438  df-fr 5487  df-we 5489  df-ord 6167 This theorem is referenced by: (None)
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