opnoncon - Mathbox for Norm Megill

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Theorem opnoncon 40531

Description: Law of contradiction for orthoposets. (chocin 32154 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 2763	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opnoncon.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		eqid 2763	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
5		opnoncon.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		opnoncon.z	. . . 4 ⊢ 0 = (0.‘𝐾)
7		eqid 2763	. . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 40505	. . 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 1449	. 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 1162	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 lecple 17479 occoc 17480 joincjn 18535 meetcmee 18536 0.cp0 18645 1.cp1 18646 OPcops 40495

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-nul 5266

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-ral 3081 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-dm 5668 df-iota 6520 df-fv 6574 df-ov 7454 df-oposet 40499

This theorem is used by: omlfh1N 40581 omlspjN 40584 atlatmstc 40642 pnonsingN 41256 lhpocnle 41339 dochnoncon 42714

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