Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalemcceb Structured version   Unicode version

Theorem dalemcceb 35810
Description: Lemma for dath 35857. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
Hypotheses
Ref Expression
da.ps0  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
da.a1  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalemcceb  |-  ( ps 
->  c  e.  ( Base `  K ) )

Proof of Theorem dalemcceb
StepHypRef Expression
1 da.ps0 . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
21dalemccea 35804 . 2  |-  ( ps 
->  c  e.  A
)
3 eqid 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 da.a1 . . 3  |-  A  =  ( Atoms `  K )
53, 4atbase 35411 . 2  |-  ( c  e.  A  ->  c  e.  ( Base `  K
) )
62, 5syl 16 1  |-  ( ps 
->  c  e.  ( Base `  K ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   Atomscatm 35385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ats 35389
This theorem is referenced by:  dalem21  35815  dalem25  35819  dalem38  35831  dalem39  35832  dalem44  35837  dalem45  35838  dalem48  35841  dalem52  35845
  Copyright terms: Public domain W3C validator