opnoncon - Mathbox for Norm Megill

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Theorem opnoncon 39654

Description: Law of contradiction for orthoposets. (chocin 31566 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 2736	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opnoncon.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		eqid 2736	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
5		opnoncon.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		opnoncon.z	. . . 4 ⊢ 0 = (0.‘𝐾)
7		eqid 2736	. . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 39628	. . 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 1424	. 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 1542 ∈ wcel 2114 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 lecple 17227 occoc 17228 joincjn 18277 meetcmee 18278 0.cp0 18387 1.cp1 18388 OPcops 39618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2708 ax-nul 5241

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-dm 5641 df-iota 6454 df-fv 6506 df-ov 7370 df-oposet 39622

This theorem is referenced by: omlfh1N 39704 omlspjN 39707 atlatmstc 39765 pnonsingN 40379 lhpocnle 40462 dochnoncon 41837

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