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

Theorem isat 38753
Description: The predicate "is an atom". (ela 32143 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 38752 . . 3 (𝐾 ∈ 𝐷 β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
65eleq2d 2815 . 2 (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯}))
7 breq2 5147 . . 3 (π‘₯ = 𝑃 β†’ ( 0 𝐢π‘₯ ↔ 0 𝐢𝑃))
87elrab 3681 . 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 1534   ∈ wcel 2099  {crab 3428   class class class wbr 5143  β€˜cfv 6543  Basecbs 17174  0.cp0 18409   β‹– ccvr 38729  Atomscatm 38730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ats 38734
This theorem is referenced by:  isat2  38754  atcvr0  38755  atbase  38756  isat3  38774  1cvrco  38940  1cvrjat  38943  ltrnatb  39605
  Copyright terms: Public domain W3C validator