Theorem dford5reg 36008

Description: Given ax-reg 9497, 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 6313	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2		zfregfr 9516	. . . 4 ⊢ E Fr 𝐴
3		df-we 5573	. . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
4	2, 3	mpbiran 715	. . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴)
5	4	anbi2i 629	. 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴))
6	1, 5	bitri 276	1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Syntax hints:   ↔ wb 207   ∧ wa 396  Tr wtr 5179   E cep 5517   Or wor 5525   Fr wfr 5568   We wwe 5570  Ord word 6309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-reg 9497

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-eprel 5518  df-fr 5571  df-we 5573  df-ord 6313

This theorem is referenced by: (None)