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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemcceb | Structured version Visualization version GIF version |
Description: Lemma for dath 36752. 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 36699 | . 2 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
3 | eqid 2818 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | da.a1 | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 36305 | . 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Atomscatm 36279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ats 36283 |
This theorem is referenced by: dalem21 36710 dalem25 36714 dalem38 36726 dalem39 36727 dalem44 36732 dalem45 36733 dalem48 36736 dalem52 36740 |
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