Theorem opexmid 37467

Description: Law of excluded middle for orthoposets. (chjo 30078 analog.) (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
opexmid.b	⊢ 𝐵 = (Base‘𝐾)
opexmid.o	⊢ ⊥ = (oc‘𝐾)
opexmid.j	⊢ ∨ = (join‘𝐾)
opexmid.u	⊢ 1 = (1.‘𝐾)

Assertion

Ref	Expression
opexmid	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )

Proof of Theorem opexmid

Step	Hyp	Ref	Expression
1		opexmid.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2736	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opexmid.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		opexmid.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		eqid 2736	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
6		eqid 2736	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		opexmid.u	. . . 4 ⊢ 1 = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 37442	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾)))
9	8	3anidm23 1420	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾)))
10	9	simp2d 1142	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   class class class wbr 5089  ‘cfv 6473  (class class class)co 7329  Basecbs 17001  lecple 17058  occoc 17059  joincjn 18118  meetcmee 18119  0.cp0 18230  1.cp1 18231  OPcops 37432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5247

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-dm 5624  df-iota 6425  df-fv 6481  df-ov 7332  df-oposet 37436

This theorem is referenced by:  dih1  39547