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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 2740 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
2 | atncvr.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atn0 39266 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
4 | 3 | 3adant3 1132 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
5 | eqid 2740 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 5, 2 | atbase 39247 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
7 | eqid 2740 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | atncvr.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
9 | 5, 7, 1, 8, 2 | atcvreq0 39272 | . . . 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 2984 | . 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 2108 ≠ wne 2946 class class class wbr 5166 ‘cfv 6575 Basecbs 17260 lecple 17320 0.cp0 18495 ⋖ ccvr 39220 Atomscatm 39221 AtLatcal 39222 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-proset 18367 df-poset 18385 df-plt 18402 df-glb 18419 df-p0 18497 df-lat 18504 df-covers 39224 df-ats 39225 df-atl 39256 |
This theorem is referenced by: (None) |
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