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Mirrors > Home > MPE Home > Th. List > Mathboxes > atncmp | Structured version Visualization version GIF version |
Description: Frequently-used variation of atcmp 36441. (Contributed by NM, 29-Jun-2012.) |
Ref | Expression |
---|---|
atcmp.l | ⊢ ≤ = (le‘𝐾) |
atcmp.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atncmp | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atcmp.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | atcmp.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atcmp 36441 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
4 | 3 | necon3bbid 3053 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 lecple 16566 Atomscatm 36393 AtLatcal 36394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-proset 17532 df-poset 17550 df-plt 17562 df-glb 17579 df-p0 17643 df-lat 17650 df-covers 36396 df-ats 36397 df-atl 36428 |
This theorem is referenced by: atnem0 36448 cvlatexchb1 36464 3dim2 36598 2dim 36600 ps-2 36608 islln3 36640 llnexatN 36651 isline4N 36907 trlnle 37316 cdleme7ga 37378 |
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