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Theorem isat 38151
Description: The predicate "is an atom". (ela 31587 analog.) (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isatom.b 𝐡 = (Baseβ€˜πΎ)
isatom.z 0 = (0.β€˜πΎ)
isatom.c 𝐢 = ( β‹– β€˜πΎ)
isatom.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
isat (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 0 𝐢𝑃)))

Proof of Theorem isat
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 isatom.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 isatom.z . . . 4 0 = (0.β€˜πΎ)
3 isatom.c . . . 4 𝐢 = ( β‹– β€˜πΎ)
4 isatom.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4pats 38150 . . 3 (𝐾 ∈ 𝐷 β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
65eleq2d 2819 . 2 (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯}))
7 breq2 5152 . . 3 (π‘₯ = 𝑃 β†’ ( 0 𝐢π‘₯ ↔ 0 𝐢𝑃))
87elrab 3683 . 2 (𝑃 ∈ {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯} ↔ (𝑃 ∈ 𝐡 ∧ 0 𝐢𝑃))
96, 8bitrdi 286 1 (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 0 𝐢𝑃)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   class class class wbr 5148  β€˜cfv 6543  Basecbs 17143  0.cp0 18375   β‹– ccvr 38127  Atomscatm 38128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ats 38132
This theorem is referenced by:  isat2  38152  atcvr0  38153  atbase  38154  isat3  38172  1cvrco  38338  1cvrjat  38341  ltrnatb  39003
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