Theorem dalemccea 33292

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
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 1038	. 2 |- ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> c e. A )
3	1, 2	sylbi 200	1 |- ( ps -> c e. A )

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:  dalemcceb  33298  dalemswapyzps  33299  dalemrotps  33300  dalemcjden  33301  dalem23  33305  dalem24  33306  dalem25  33307  dalem27  33308  dalem28  33309  dalem38  33319  dalem39  33320  dalem44  33325  dalem51  33332  dalem56  33337