![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > opnoncon | Structured version Visualization version GIF version |
Description: Law of contradiction for orthoposets. (chocin 29278 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 2798 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opnoncon.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | eqid 2798 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | opnoncon.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
6 | opnoncon.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
7 | eqid 2798 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 36478 | . . 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 1418 | . 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 1141 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 occoc 16565 joincjn 17546 meetcmee 17547 0.cp0 17639 1.cp1 17640 OPcops 36468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 df-oposet 36472 |
This theorem is referenced by: omlfh1N 36554 omlspjN 36557 atlatmstc 36615 pnonsingN 37229 lhpocnle 37312 dochnoncon 38687 |
Copyright terms: Public domain | W3C validator |