Theorem dford5reg 36238

Description: Given ax-reg 9553, 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 6363	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2		zfregfr 9572	. . . 4 ⊢ E Fr 𝐴
3		df-we 5616	. . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
4	2, 3	mpbiran 721	. . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴)
5	4	anbi2i 634	. 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
6	1, 5	bitri 278	1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400  Tr wtr 5217   E cep 5560   Or wor 5568   Fr wfr 5611   We wwe 5613  Ord word 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404  ax-reg 9553

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-eprel 5561  df-fr 5614  df-we 5616  df-ord 6363

This theorem is referenced by: (None)