Theorem dalem-ccly 35806

Description: Lemma for dath 35857. 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 1010	1 |- ( ps -> -. c .<_ Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   e. wcel 1823   =/= wne 2649   class class class wbr 4439  (class class class)co 6270

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 185  df-an 369  df-3an 973

This theorem is referenced by:  dalemswapyzps  35811  dalemrotps  35812  dalem21  35815  dalem23  35817  dalem24  35818  dalem39  35832  dalem48  35841