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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemcceb | Structured version Visualization version GIF version |
Description: Lemma for dath 35022. 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 34969 | . 2 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
3 | eqid 2622 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | da.a1 | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 34576 | . 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Atomscatm 34550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ats 34554 |
This theorem is referenced by: dalem21 34980 dalem25 34984 dalem38 34996 dalem39 34997 dalem44 35002 dalem45 35003 dalem48 35006 dalem52 35010 |
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