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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddasslem1 | Structured version Visualization version GIF version |
Description: Lemma for paddass 37848. (Contributed by NM, 8-Jan-2012.) |
Ref | Expression |
---|---|
paddasslem.l | ⊢ ≤ = (le‘𝐾) |
paddasslem.j | ⊢ ∨ = (join‘𝐾) |
paddasslem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
paddasslem1 | ⊢ (((𝐾 ∈ HL ∧ (𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) ∧ ¬ 𝑟 ≤ (𝑥 ∨ 𝑦)) → ¬ 𝑥 ≤ (𝑟 ∨ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | paddasslem.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | paddasslem.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | paddasslem.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 2, 3 | hlatexch2 37406 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → (𝑥 ≤ (𝑟 ∨ 𝑦) → 𝑟 ≤ (𝑥 ∨ 𝑦))) |
5 | 4 | con3dimp 409 | 1 ⊢ (((𝐾 ∈ HL ∧ (𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) ∧ ¬ 𝑟 ≤ (𝑥 ∨ 𝑦)) → ¬ 𝑥 ≤ (𝑟 ∨ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 lecple 16967 joincjn 18027 Atomscatm 37273 HLchlt 37360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-lat 18148 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 |
This theorem is referenced by: paddasslem4 37833 |
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