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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnoncon | Structured version Visualization version GIF version |
Description: Law of contradiction for orthoposets. (chocin 28482 analog.) (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opnoncon.b | ⊢ 𝐵 = (Base‘𝐾) |
opnoncon.o | ⊢ ⊥ = (oc‘𝐾) |
opnoncon.m | ⊢ ∧ = (meet‘𝐾) |
opnoncon.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
opnoncon | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opnoncon.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2651 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opnoncon.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | eqid 2651 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | opnoncon.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
6 | opnoncon.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
7 | eqid 2651 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 34787 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )) |
9 | 8 | 3anidm23 1425 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )) |
10 | 9 | simp3d 1095 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 lecple 15995 occoc 15996 joincjn 16991 meetcmee 16992 0.cp0 17084 1.cp1 17085 OPcops 34777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-dm 5153 df-iota 5889 df-fv 5934 df-ov 6693 df-oposet 34781 |
This theorem is referenced by: omlfh1N 34863 omlspjN 34866 atlatmstc 34924 pnonsingN 35537 lhpocnle 35620 dochnoncon 36997 |
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