Theorem opexmid 40371

Description: Law of excluded middle for orthoposets. (chjo 32124 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 2760	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opexmid.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		opexmid.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		eqid 2760	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
6		eqid 2760	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		opexmid.u	. . . 4 ⊢ 1 = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 40346	. . 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 1448	. 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 1161	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  lecple 17442  occoc 17443  joincjn 18492  meetcmee 18493  0.cp0 18602  1.cp1 18603  OPcops 40336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-dm 5660  df-iota 6496  df-fv 6549  df-ov 7425  df-oposet 40340

This theorem is used by:  dih1  42450