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Theorem dalemcceb 36861
Description: Lemma for dath 36908. 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 36855 . 2 (𝜓𝑐𝐴)
3 eqid 2820 . . 3 (Base‘𝐾) = (Base‘𝐾)
4 da.a1 . . 3 𝐴 = (Atoms‘𝐾)
53, 4atbase 36461 . 2 (𝑐𝐴𝑐 ∈ (Base‘𝐾))
62, 5syl 17 1 (𝜓𝑐 ∈ (Base‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  Atomscatm 36435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-ats 36439
This theorem is referenced by:  dalem21  36866  dalem25  36870  dalem38  36882  dalem39  36883  dalem44  36888  dalem45  36889  dalem48  36892  dalem52  36896
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