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Mirrors > Home > MPE Home > Th. List > Mathboxes > pexmidlem7N | Structured version Visualization version GIF version |
Description: Lemma for pexmidN 36045. Contradict pexmidlem6N 36051. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pexmidlem.l | ⊢ ≤ = (le‘𝐾) |
pexmidlem.j | ⊢ ∨ = (join‘𝐾) |
pexmidlem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pexmidlem.p | ⊢ + = (+𝑃‘𝐾) |
pexmidlem.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
pexmidlem.m | ⊢ 𝑀 = (𝑋 + {𝑝}) |
Ref | Expression |
---|---|
pexmidlem7N | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 ≠ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1248 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝐾 ∈ HL) | |
2 | simpl3 1252 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑝 ∈ 𝐴) | |
3 | 2 | snssd 4559 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → {𝑝} ⊆ 𝐴) |
4 | simpl2 1250 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑋 ⊆ 𝐴) | |
5 | pexmidlem.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | pexmidlem.p | . . . . . 6 ⊢ + = (+𝑃‘𝐾) | |
7 | 5, 6 | sspadd2 35892 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ {𝑝} ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) → {𝑝} ⊆ (𝑋 + {𝑝})) |
8 | 1, 3, 4, 7 | syl3anc 1496 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → {𝑝} ⊆ (𝑋 + {𝑝})) |
9 | vex 3418 | . . . . 5 ⊢ 𝑝 ∈ V | |
10 | 9 | snss 4536 | . . . 4 ⊢ (𝑝 ∈ (𝑋 + {𝑝}) ↔ {𝑝} ⊆ (𝑋 + {𝑝})) |
11 | 8, 10 | sylibr 226 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑝 ∈ (𝑋 + {𝑝})) |
12 | pexmidlem.m | . . 3 ⊢ 𝑀 = (𝑋 + {𝑝}) | |
13 | 11, 12 | syl6eleqr 2918 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑝 ∈ 𝑀) |
14 | pexmidlem.o | . . . . . 6 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
15 | 5, 14 | polssatN 35984 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
16 | 1, 4, 15 | syl2anc 581 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
17 | 5, 6 | sspadd1 35891 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + ( ⊥ ‘𝑋))) |
18 | 1, 4, 16, 17 | syl3anc 1496 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑋 ⊆ (𝑋 + ( ⊥ ‘𝑋))) |
19 | simpr3 1258 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) | |
20 | 18, 19 | ssneldd 3831 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ¬ 𝑝 ∈ 𝑋) |
21 | nelne1 3096 | . 2 ⊢ ((𝑝 ∈ 𝑀 ∧ ¬ 𝑝 ∈ 𝑋) → 𝑀 ≠ 𝑋) | |
22 | 13, 20, 21 | syl2anc 581 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 ≠ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ⊆ wss 3799 ∅c0 4145 {csn 4398 ‘cfv 6124 (class class class)co 6906 lecple 16313 joincjn 17298 Atomscatm 35339 HLchlt 35426 +𝑃cpadd 35871 ⊥𝑃cpolN 35978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-riotaBAD 35029 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-undef 7665 df-proset 17282 df-poset 17300 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p1 17394 df-lat 17400 df-clat 17462 df-oposet 35252 df-ol 35254 df-oml 35255 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-polarityN 35979 |
This theorem is referenced by: pexmidlem8N 36053 |
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