Metamath Proof Explorer


Theorem dalem-ccly

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

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

Proof

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