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Theorem dalemcceb 35759
Description: Lemma for dath 35806. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
Hypotheses
Ref Expression
da.ps0 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
da.a1 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dalemcceb (𝜓𝑐 ∈ (Base‘𝐾))

Proof of Theorem dalemcceb
StepHypRef Expression
1 da.ps0 . . 3 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
21dalemccea 35753 . 2 (𝜓𝑐𝐴)
3 eqid 2825 . . 3 (Base‘𝐾) = (Base‘𝐾)
4 da.a1 . . 3 𝐴 = (Atoms‘𝐾)
53, 4atbase 35359 . 2 (𝑐𝐴𝑐 ∈ (Base‘𝐾))
62, 5syl 17 1 (𝜓𝑐 ∈ (Base‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  Atomscatm 35333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ats 35337
This theorem is referenced by:  dalem21  35764  dalem25  35768  dalem38  35780  dalem39  35781  dalem44  35786  dalem45  35787  dalem48  35790  dalem52  35794
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