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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpexle2 | Structured version Visualization version GIF version |
Description: There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.) |
Ref | Expression |
---|---|
lhpex1.l | ⊢ ≤ = (le‘𝐾) |
lhpex1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpex1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpexle2 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpex1.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | lhpex1.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | lhpex1.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhpexle1 37708 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋)) |
5 | 1, 2, 3 | lhpexle1 37708 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌)) |
6 | 5 | adantr 484 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌)) |
7 | 1, 2, 3 | lhpexle2lem 37709 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊) ∧ (𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋)) |
8 | 7 | 3expa 1120 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋)) |
9 | 6, 8 | lhpexle1lem 37707 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋)) |
10 | 3ancomb 1101 | . . . 4 ⊢ ((𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋) ↔ (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌)) | |
11 | 10 | rexbii 3160 | . . 3 ⊢ (∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋) ↔ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌)) |
12 | 9, 11 | sylib 221 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌)) |
13 | 4, 12 | lhpexle1lem 37707 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∃wrex 3052 class class class wbr 5039 ‘cfv 6358 lecple 16756 Atomscatm 36963 HLchlt 37050 LHypclh 37684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-lhyp 37688 |
This theorem is referenced by: lhpexle3lem 37711 lhpexle3 37712 cdlemj3 38523 |
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