Theorem poleloe 5167

Description: Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
poleloe	|- ( B e. V -> ( A ( R u. _I ) B <-> ( A R B \/ A = B ) ) )

Proof of Theorem poleloe

Step	Hyp	Ref	Expression
1		brun 4166	. 2 |- ( A ( R u. _I ) B <-> ( A R B \/ A _I B ) )
2		ideqg 4911	. . 3 |- ( B e. V -> ( A _I B <-> A = B ) )
3	2	orbi2d 798	. 2 |- ( B e. V -> ( ( A R B \/ A _I B ) <-> ( A R B \/ A = B ) ) )
4	1, 3	bitrid 192	1 |- ( B e. V -> ( A ( R u. _I ) B <-> ( A R B \/ A = B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   u. cun 3212   class class class wbr 4114   _I cid 4414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761

This theorem is referenced by:  poltletr  5168