Theorem opexmid 39201

Description: Law of excluded middle for orthoposets. (chjo 31557 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 39176	. . 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 1421	. 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 1143	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1538   ∈ wcel 2107   class class class wbr 5149  ‘cfv 6566  (class class class)co 7435  Basecbs 17251  lecple 17311  occoc 17312  joincjn 18375  meetcmee 18376  0.cp0 18487  1.cp1 18488  OPcops 39166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5313

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-dm 5700  df-iota 6519  df-fv 6574  df-ov 7438  df-oposet 39170

This theorem is referenced by:  dih1  41281