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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemcceb | Structured version Visualization version GIF version |
Description: Lemma for dath 35806. 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 35753 | . 2 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
3 | eqid 2825 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | da.a1 | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 35359 | . 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 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 Atomscatm 35333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-ats 35337 |
This theorem is referenced by: dalem21 35764 dalem25 35768 dalem38 35780 dalem39 35781 dalem44 35786 dalem45 35787 dalem48 35790 dalem52 35794 |
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