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Mirrors > Home > MPE Home > Th. List > Mathboxes > atncmp | Structured version Visualization version GIF version |
Description: Frequently-used variation of atcmp 36462. (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 36462 | . 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 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 lecple 16572 Atomscatm 36414 AtLatcal 36415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-proset 17538 df-poset 17556 df-plt 17568 df-glb 17585 df-p0 17649 df-lat 17656 df-covers 36417 df-ats 36418 df-atl 36449 |
This theorem is referenced by: atnem0 36469 cvlatexchb1 36485 3dim2 36619 2dim 36621 ps-2 36629 islln3 36661 llnexatN 36672 isline4N 36928 trlnle 37337 cdleme7ga 37399 |
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