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Theorem dford5reg 32013
Description: Given ax-reg 8664, 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 5887 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 8676 . . . 4 E Fr 𝐴
3 df-we 5227 . . . 4 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
42, 3mpbiran 991 . . 3 ( E We 𝐴 ↔ E Or 𝐴)
54anbi2i 732 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
61, 5bitri 264 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  Tr wtr 4904   E cep 5178   Or wor 5186   Fr wfr 5222   We wwe 5224  Ord word 5883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-reg 8664
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-eprel 5179  df-fr 5225  df-we 5227  df-ord 5887
This theorem is referenced by: (None)
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