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Theorem isat 38650
Description: The predicate "is an atom". (ela 32064 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 38649 . . 3 (𝐾 ∈ 𝐷 β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
65eleq2d 2811 . 2 (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯}))
7 breq2 5143 . . 3 (π‘₯ = 𝑃 β†’ ( 0 𝐢π‘₯ ↔ 0 𝐢𝑃))
87elrab 3676 . 2 (𝑃 ∈ {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯} ↔ (𝑃 ∈ 𝐡 ∧ 0 𝐢𝑃))
96, 8bitrdi 287 1 (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 0 𝐢𝑃)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424   class class class wbr 5139  β€˜cfv 6534  Basecbs 17145  0.cp0 18380   β‹– ccvr 38626  Atomscatm 38627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ats 38631
This theorem is referenced by:  isat2  38651  atcvr0  38652  atbase  38653  isat3  38671  1cvrco  38837  1cvrjat  38840  ltrnatb  39502
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