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Theorem dford5reg 34396
Description: Given ax-reg 9535, 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 6325 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 9548 . . . 4 E Fr 𝐴
3 df-we 5595 . . . 4 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
42, 3mpbiran 708 . . 3 ( E We 𝐴 ↔ E Or 𝐴)
54anbi2i 624 . 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 205  wa 397  Tr wtr 5227   E cep 5541   Or wor 5549   Fr wfr 5590   We wwe 5592  Ord word 6321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-reg 9535
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-eprel 5542  df-fr 5593  df-we 5595  df-ord 6325
This theorem is referenced by: (None)
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