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Mirrors > Home > MPE Home > Th. List > Mathboxes > atncmp | Structured version Visualization version GIF version |
Description: Frequently-used variation of atcmp 38644. (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 38644 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
4 | 3 | necon3bbid 2977 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 lecple 17211 Atomscatm 38596 AtLatcal 38597 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-proset 18258 df-poset 18276 df-plt 18293 df-glb 18310 df-p0 18388 df-lat 18395 df-covers 38599 df-ats 38600 df-atl 38631 |
This theorem is referenced by: atnem0 38651 cvlatexchb1 38667 3dim2 38802 2dim 38804 ps-2 38812 islln3 38844 llnexatN 38855 isline4N 39111 trlnle 39520 cdleme7ga 39582 |
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