Theorem dalemccea 33635

Description: Lemma for dath 33688. 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
dalemccea	|- ( ps -> c e. A )

Proof of Theorem dalemccea

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		simp1l 1012	. 2 |- ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> c e. A )
3	1, 2	sylbi 195	1 |- ( ps -> c e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   e. wcel 1758   =/= wne 2644   class class class wbr 4392  (class class class)co 6192

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 371  df-3an 967

This theorem is referenced by:  dalemcceb  33641  dalemswapyzps  33642  dalemrotps  33643  dalemcjden  33644  dalem23  33648  dalem24  33649  dalem25  33650  dalem27  33651  dalem28  33652  dalem38  33662  dalem39  33663  dalem44  33668  dalem51  33675  dalem56  33680