Theorem opnoncon

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

	Ref	Expression
Hypotheses	opnoncon.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	opnoncon.o	⊢ ⊥ = ( oc ‘ 𝐾 )
	opnoncon.m	⊢ ∧ = ( meet ‘ 𝐾 )
	opnoncon.z	⊢ 0 = ( 0. ‘ 𝐾 )
Assertion	opnoncon	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ ( ⊥ ‘ 𝑋 ) ) = 0 )

Proof

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