Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > opexmid | Structured version Visualization version GIF version |
Description: Law of excluded middle for orthoposets. (chjo 29295 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 2824 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opexmid.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | opexmid.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | eqid 2824 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2824 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | opexmid.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 36322 | . . 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 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 lecple 16575 occoc 16576 joincjn 17557 meetcmee 17558 0.cp0 17650 1.cp1 17651 OPcops 36312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 df-ov 7162 df-oposet 36316 |
This theorem is referenced by: dih1 38426 |
Copyright terms: Public domain | W3C validator |