Theorem opexmid 40530

Description: Law of excluded middle for orthoposets. (chjo 32174 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 2763	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opexmid.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		opexmid.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		eqid 2763	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
6		eqid 2763	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		opexmid.u	. . . 4 ⊢ 1 = (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 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾)))
9	8	3anidm23 1449	. 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 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:  dih1  42609