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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 32368 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 39267 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
6 | 5 | eleq2d 2825 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})) |
7 | breq2 5152 | . . 3 ⊢ (𝑥 = 𝑃 → ( 0 𝐶𝑥 ↔ 0 𝐶𝑃)) | |
8 | 7 | elrab 3695 | . 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃)) |
9 | 6, 8 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 0.cp0 18481 ⋖ ccvr 39244 Atomscatm 39245 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ats 39249 |
This theorem is referenced by: isat2 39269 atcvr0 39270 atbase 39271 isat3 39289 1cvrco 39455 1cvrjat 39458 ltrnatb 40120 |
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