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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 28582 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 33590 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
6 | 5 | eleq2d 2673 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})) |
7 | breq2 4587 | . . 3 ⊢ (𝑥 = 𝑃 → ( 0 𝐶𝑥 ↔ 0 𝐶𝑃)) | |
8 | 7 | elrab 3331 | . 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃)) |
9 | 6, 8 | syl6bb 275 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 0.cp0 16860 ⋖ ccvr 33567 Atomscatm 33568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ats 33572 |
This theorem is referenced by: isat2 33592 atcvr0 33593 atbase 33594 isat3 33612 1cvrco 33776 1cvrjat 33779 ltrnatb 34441 |
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