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Mirrors > Home > MPE Home > Th. List > Mathboxes > atncvrN | Structured version Visualization version GIF version |
Description: Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atncvr.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
atncvr.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atncvrN | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃𝐶𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
2 | atncvr.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atn0 36324 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
4 | 3 | 3adant3 1124 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
5 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 5, 2 | atbase 36305 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
7 | eqid 2818 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | atncvr.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
9 | 5, 7, 1, 8, 2 | atcvreq0 36330 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴) → (𝑃𝐶𝑄 ↔ 𝑃 = (0.‘𝐾))) |
10 | 6, 9 | syl3an2 1156 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃𝐶𝑄 ↔ 𝑃 = (0.‘𝐾))) |
11 | 10 | necon3bbid 3050 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃𝐶𝑄 ↔ 𝑃 ≠ (0.‘𝐾))) |
12 | 4, 11 | mpbird 258 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃𝐶𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 0.cp0 17635 ⋖ ccvr 36278 Atomscatm 36279 AtLatcal 36280 |
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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
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-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 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-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-proset 17526 df-poset 17544 df-plt 17556 df-glb 17573 df-p0 17637 df-lat 17644 df-covers 36282 df-ats 36283 df-atl 36314 |
This theorem is referenced by: (None) |
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