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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemcceb | Structured version Visualization version GIF version | ||
| Description: Lemma for dath 39738. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
| Ref | Expression |
|---|---|
| da.ps0 | ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| da.a1 | ⊢ 𝐴 = (Atoms‘𝐾) |
| Ref | Expression |
|---|---|
| dalemcceb | ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | da.ps0 | . . 3 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
| 2 | 1 | dalemccea 39685 | . 2 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
| 3 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 4 | da.a1 | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | 3, 4 | atbase 39290 | . 2 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ (Base‘𝐾)) |
| 6 | 2, 5 | syl 17 | 1 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 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: dalem21 39696 dalem25 39700 dalem38 39712 dalem39 39713 dalem44 39718 dalem45 39719 dalem48 39722 dalem52 39726 |
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