Theorem dford5reg 35955

Description: Given ax-reg 9501, 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 6321	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2		zfregfr 9517	. . . 4 ⊢ E Fr 𝐴
3		df-we 5580	. . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
4	2, 3	mpbiran 710	. . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴)
5	4	anbi2i 624	. 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
6	1, 5	bitri 275	1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395  Tr wtr 5206   E cep 5524   Or wor 5532   Fr wfr 5575   We wwe 5577  Ord word 6317

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-eprel 5525  df-fr 5578  df-we 5580  df-ord 6321

This theorem is referenced by: (None)