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Mathbox for Norm Megill |
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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 2734 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
2 | atncvr.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atn0 39213 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
4 | 3 | 3adant3 1132 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
5 | eqid 2734 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 5, 2 | atbase 39194 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
7 | eqid 2734 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | atncvr.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
9 | 5, 7, 1, 8, 2 | atcvreq0 39219 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴) → (𝑃𝐶𝑄 ↔ 𝑃 = (0.‘𝐾))) |
10 | 6, 9 | syl3an2 1164 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃𝐶𝑄 ↔ 𝑃 = (0.‘𝐾))) |
11 | 10 | necon3bbid 2980 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃𝐶𝑄 ↔ 𝑃 ≠ (0.‘𝐾))) |
12 | 4, 11 | mpbird 257 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃𝐶𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 class class class wbr 5169 ‘cfv 6572 Basecbs 17253 lecple 17313 0.cp0 18488 ⋖ ccvr 39167 Atomscatm 39168 AtLatcal 39169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-proset 18360 df-poset 18378 df-plt 18395 df-glb 18412 df-p0 18490 df-lat 18497 df-covers 39171 df-ats 39172 df-atl 39203 |
This theorem is referenced by: (None) |
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