Theorem dford5reg 35777

Description: Given ax-reg 9552, 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 6338	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2		zfregfr 9565	. . . 4 ⊢ E Fr 𝐴
3		df-we 5596	. . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴))
4	2, 3	mpbiran 709	. . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴)
5	4	anbi2i 623	. 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 5217   E cep 5540   Or wor 5548   Fr wfr 5591   We wwe 5593  Ord word 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-reg 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-eprel 5541  df-fr 5594  df-we 5596  df-ord 6338

This theorem is referenced by: (None)