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Mirrors > Home > MPE Home > Th. List > Mathboxes > opexmid | Structured version Visualization version GIF version |
Description: Law of excluded middle for orthoposets. (chjo 29292 analog.) (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opexmid.b | ⊢ 𝐵 = (Base‘𝐾) |
opexmid.o | ⊢ ⊥ = (oc‘𝐾) |
opexmid.j | ⊢ ∨ = (join‘𝐾) |
opexmid.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
opexmid | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opexmid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opexmid.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | opexmid.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | eqid 2821 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2821 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | opexmid.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 36333 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾))) |
9 | 8 | 3anidm23 1417 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾))) |
10 | 9 | simp2d 1139 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 lecple 16572 occoc 16573 joincjn 17554 meetcmee 17555 0.cp0 17647 1.cp1 17648 OPcops 36323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-dm 5565 df-iota 6314 df-fv 6363 df-ov 7159 df-oposet 36327 |
This theorem is referenced by: dih1 38437 |
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