Theorem isat 35362

Description: The predicate "is an atom". (ela 29754 analog.) (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isatom.b	⊢ 𝐵 = (Base‘𝐾)
isatom.z	⊢ 0 = (0.‘𝐾)
isatom.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
isatom.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
isat	⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃)))

Proof of Theorem isat

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isatom.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		isatom.z	. . . 4 ⊢ 0 = (0.‘𝐾)
3		isatom.c	. . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾)
4		isatom.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	pats 35361	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
6	5	eleq2d 2893	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}))
7		breq2 4878	. . 3 ⊢ (𝑥 = 𝑃 → ( 0 𝐶𝑥 ↔ 0 𝐶𝑃))
8	7	elrab 3586	. 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))
9	6, 8	syl6bb 279	1 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3122   class class class wbr 4874  ‘cfv 6124  Basecbs 16223  0.cp0 17391   ⋖ ccvr 35338  Atomscatm 35339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-ats 35343

This theorem is referenced by:  isat2  35363  atcvr0  35364  atbase  35365  isat3  35383  1cvrco  35548  1cvrjat  35551  ltrnatb  36213