Theorem dalemccea 35141

Description: Lemma for dath 35194. 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 1021	. 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 974   e. wcel 1804   =/= wne 2638   class class class wbr 4437  (class class class)co 6281

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 976

This theorem is referenced by:  dalemcceb  35147  dalemswapyzps  35148  dalemrotps  35149  dalemcjden  35150  dalem23  35154  dalem24  35155  dalem25  35156  dalem27  35157  dalem28  35158  dalem38  35168  dalem39  35169  dalem44  35174  dalem51  35181  dalem56  35186