Theorem dford5reg 31687

Description: Given ax-reg 8497, an ordinal is a transitive class totally ordered by epsilon. (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 5726	. 2 |- ( Ord A <-> ( Tr A /\ _E We A ) )
2		zfregfr 8509	. . . 4 |- _E Fr A
3		df-we 5075	. . . 4 |- ( _E We A <-> ( _E Fr A /\ _E Or A ) )
4	2, 3	mpbiran 953	. . 3 |- ( _E We A <-> _E Or A )
5	4	anbi2i 730	. 2 |- ( ( Tr A /\ _E We A ) <-> ( Tr A /\ _E Or A ) )
6	1, 5	bitri 264	1 |- ( Ord A <-> ( Tr A /\ _E Or A ) )

Syntax hints:   <-> wb 196   /\ wa 384   Tr wtr 4752   _E cep 5028   Or wor 5034   Fr wfr 5070   We wwe 5072   Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by: (None)