|   | Mathbox for Norm Megill | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > isat | Structured version Visualization version GIF version | ||
| Description: The predicate "is an atom". (ela 32358 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 39286 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) | 
| 6 | 5 | eleq2d 2827 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})) | 
| 7 | breq2 5147 | . . 3 ⊢ (𝑥 = 𝑃 → ( 0 𝐶𝑥 ↔ 0 𝐶𝑃)) | |
| 8 | 7 | elrab 3692 | . 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 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 0.cp0 18468 ⋖ ccvr 39263 Atomscatm 39264 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ats 39268 | 
| This theorem is referenced by: isat2 39288 atcvr0 39289 atbase 39290 isat3 39308 1cvrco 39474 1cvrjat 39477 ltrnatb 40139 | 
| Copyright terms: Public domain | W3C validator |