Theorem dford5reg 33737

Description: Given ax-reg 9312, 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

Step	Hyp	Ref	Expression
1		df-ord 6266	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2		zfregfr 9324	. . . 4 ⊢ E Fr 𝐴
3		df-we 5545	. . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
4	2, 3	mpbiran 705	. . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴)
5	4	anbi2i 622	. 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
6	1, 5	bitri 274	1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 395  Tr wtr 5195   E cep 5493   Or wor 5501   Fr wfr 5540   We wwe 5542  Ord word 6262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-reg 9312

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-eprel 5494  df-fr 5543  df-we 5545  df-ord 6266

This theorem is referenced by: (None)