Theorem poleloe 5182

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 4177	. 2 |- ( A ( R u. _I ) B <-> ( A R B \/ A _I B ) )
2		ideqg 4926	. . 3 |- ( B e. V -> ( A _I B <-> A = B ) )
3	2	orbi2d 802	. 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 720   = wceq 1402   e. wcel 2209   u. cun 3218   class class class wbr 4125   _I cid 4428

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776

This theorem is referenced by:  poltletr  5183