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Mirrors > Home > MPE Home > Th. List > Mathboxes > isat | Structured version Visualization version GIF version |
Description: The predicate "is an atom". (ela 30110 analog.) (Contributed by NM, 18-Sep-2011.) |
Ref | Expression |
---|---|
isatom.b | ⊢ 𝐵 = (Base‘𝐾) |
isatom.z | ⊢ 0 = (0.‘𝐾) |
isatom.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
isatom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
isat | ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) |
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 36415 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
6 | 5 | eleq2d 2898 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})) |
7 | breq2 5063 | . . 3 ⊢ (𝑥 = 𝑃 → ( 0 𝐶𝑥 ↔ 0 𝐶𝑃)) | |
8 | 7 | elrab 3680 | . 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃)) |
9 | 6, 8 | syl6bb 289 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 0.cp0 17641 ⋖ ccvr 36392 Atomscatm 36393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ats 36397 |
This theorem is referenced by: isat2 36417 atcvr0 36418 atbase 36419 isat3 36437 1cvrco 36602 1cvrjat 36605 ltrnatb 37267 |
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