Theorem dalem-ccly 33294

Description: Lemma for dath 33345. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Hypothesis

Ref	Expression
da.ps0	|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )

Assertion

Ref	Expression
dalem-ccly	|- ( ps -> -. c .<_ Y )

Proof of Theorem dalem-ccly

Step	Hyp	Ref	Expression
1		da.ps0	. 2 |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
2	1	simp2bi 1030	1 |- ( ps -> -. c .<_ Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   e. wcel 1897   =/= wne 2632   class class class wbr 4415  (class class class)co 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993

This theorem is referenced by:  dalemswapyzps  33299  dalemrotps  33300  dalem21  33303  dalem23  33305  dalem24  33306  dalem39  33320  dalem48  33329