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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 28374 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 2622 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opexmid.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
4 | opexmid.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | eqid 2622 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2622 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | opexmid.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 34469 | . . 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 1385 | . 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 1074 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 occoc 15949 joincjn 16944 meetcmee 16945 0.cp0 17037 1.cp1 17038 OPcops 34459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oposet 34463 |
This theorem is referenced by: dih1 36575 |
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