Theorem opexmid 38588

Description: Law of excluded middle for orthoposets. (chjo 31273 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 2726	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opexmid.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		opexmid.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		eqid 2726	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
6		eqid 2726	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		opexmid.u	. . . 4 ⊢ 1 = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 38563	. . 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 1418	. 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 1140	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  ‘cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  occoc 17212  joincjn 18274  meetcmee 18275  0.cp0 18386  1.cp1 18387  OPcops 38553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6488  df-fv 6544  df-ov 7407  df-oposet 38557

This theorem is referenced by:  dih1  40668