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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 29199 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 2818 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opnoncon.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | eqid 2818 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | opnoncon.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
6 | opnoncon.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
7 | eqid 2818 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 36198 | . . 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 1413 | . 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 1136 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 occoc 16561 joincjn 17542 meetcmee 17543 0.cp0 17635 1.cp1 17636 OPcops 36188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 df-oposet 36192 |
This theorem is referenced by: omlfh1N 36274 omlspjN 36277 atlatmstc 36335 pnonsingN 36949 lhpocnle 37032 dochnoncon 38407 |
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