Theorem dalemccea 34969

Description: Lemma for dath 35022. 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 1085	. 2 |- ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> c e. A )
3	1, 2	sylbi 207	1 |- ( ps -> c e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  dalemcceb  34975  dalemswapyzps  34976  dalemrotps  34977  dalemcjden  34978  dalem23  34982  dalem24  34983  dalem25  34984  dalem27  34985  dalem28  34986  dalem38  34996  dalem39  34997  dalem44  35002  dalem51  35009  dalem56  35014