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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atncmp | Structured version Visualization version GIF version | ||
| Description: Frequently-used variation of atcmp 39312. (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 39312 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
| 4 | 3 | necon3bbid 2978 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 lecple 17304 Atomscatm 39264 AtLatcal 39265 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-proset 18340 df-poset 18359 df-plt 18375 df-glb 18392 df-p0 18470 df-lat 18477 df-covers 39267 df-ats 39268 df-atl 39299 |
| This theorem is referenced by: atnem0 39319 cvlatexchb1 39335 3dim2 39470 2dim 39472 ps-2 39480 islln3 39512 llnexatN 39523 isline4N 39779 trlnle 40188 cdleme7ga 40250 |
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