opnoncon - Mathbox for Norm Megill

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Theorem opnoncon 39209

Description: Law of contradiction for orthoposets. (chocin 31514 analog.) (Contributed by NM, 13-Sep-2011.)

**Hypotheses**
Ref	Expression
opnoncon.b	⊢ 𝐵 = (Base‘𝐾)
opnoncon.o	⊢ ⊥ = (oc‘𝐾)
opnoncon.m	⊢ ∧ = (meet‘𝐾)
opnoncon.z	⊢ 0 = (0.‘𝐾)

**Assertion**
Ref	Expression
opnoncon	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )

**Proof of Theorem opnoncon**
Step	Hyp	Ref	Expression
1		opnoncon.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2737	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opnoncon.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		eqid 2737	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
5		opnoncon.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		opnoncon.z	. . . 4 ⊢ 0 = (0.‘𝐾)
7		eqid 2737	. . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 39183	. . 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 1423	. 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 1145	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 0.cp0 18468 1.cp1 18469 OPcops 39173

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-oposet 39177

This theorem is referenced by: omlfh1N 39259 omlspjN 39262 atlatmstc 39320 pnonsingN 39935 lhpocnle 40018 dochnoncon 41393

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