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Mirrors > Home > MPE Home > Th. List > Mathboxes > opcon1b | Structured version Visualization version GIF version |
Description: Orthocomplement contraposition law. (negcon1 11513 analog.) (Contributed by NM, 24-Jan-2012.) |
Ref | Expression |
---|---|
opoccl.b | ⊢ 𝐵 = (Base‘𝐾) |
opoccl.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opcon1b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opoccl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | opoccl.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
3 | 1, 2 | opcon2b 38579 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋))) |
4 | eqcom 2733 | . . 3 ⊢ (( ⊥ ‘𝑌) = 𝑋 ↔ 𝑋 = ( ⊥ ‘𝑌)) | |
5 | eqcom 2733 | . . 3 ⊢ (( ⊥ ‘𝑋) = 𝑌 ↔ 𝑌 = ( ⊥ ‘𝑋)) | |
6 | 3, 4, 5 | 3bitr4g 314 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = 𝑌)) |
7 | 6 | bicomd 222 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 Basecbs 17150 occoc 17211 OPcops 38554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-dm 5679 df-iota 6488 df-fv 6544 df-ov 7407 df-oposet 38558 |
This theorem is referenced by: opoc0 38585 |
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