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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 27738 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 2610 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opnoncon.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | eqid 2610 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | opnoncon.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
6 | opnoncon.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
7 | eqid 2610 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 33487 | . . 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 1377 | . 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 1068 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 occoc 15776 joincjn 16767 meetcmee 16768 0.cp0 16860 1.cp1 16861 OPcops 33477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 df-oposet 33481 |
This theorem is referenced by: omlfh1N 33563 omlspjN 33566 atlatmstc 33624 pnonsingN 34237 lhpocnle 34320 dochnoncon 35698 |
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