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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isat2 | Structured version Visualization version GIF version |
Description: The predicate "is an atom". (elatcv0 32223 analog.) (Contributed by NM, 18-Jun-2012.) |
Ref | Expression |
---|---|
isatom.b | ⊢ 𝐵 = (Base‘𝐾) |
isatom.z | ⊢ 0 = (0.‘𝐾) |
isatom.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
isatom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
isat2 | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ 0 𝐶𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isatom.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | isatom.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | isatom.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | isatom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | isat 38888 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) |
6 | 5 | baibd 538 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ 0 𝐶𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 Basecbs 17183 0.cp0 18418 ⋖ ccvr 38864 Atomscatm 38865 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ats 38869 |
This theorem is referenced by: llncvrlpln 39161 lplncvrlvol 39219 lhpm0atN 39632 |
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