Theorem opnoncon 36346

Description: Law of contradiction for orthoposets. (chocin 29274 analog.) (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
opnoncon.b	⊢ 𝐵 = (Base‘𝐾)
opnoncon.o	⊢ ⊥ = (oc‘𝐾)
opnoncon.m	⊢ ∧ = (meet‘𝐾)
opnoncon.z	⊢ 0 = (0.‘𝐾)

Assertion

Ref	Expression
opnoncon	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )

Proof of Theorem opnoncon

Step	Hyp	Ref	Expression
1		opnoncon.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2823	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		opnoncon.o	. . . 4 ⊢ ⊥ = (oc‘𝐾)
4		eqid 2823	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
5		opnoncon.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		opnoncon.z	. . . 4 ⊢ 0 = (0.‘𝐾)
7		eqid 2823	. . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 36320	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
9	8	3anidm23 1417	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
10	9	simp3d 1140	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  occoc 16575  joincjn 17556  meetcmee 17557  0.cp0 17649  1.cp1 17650  OPcops 36310

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-dm 5567  df-iota 6316  df-fv 6365  df-ov 7161  df-oposet 36314

This theorem is referenced by:  omlfh1N  36396  omlspjN  36399  atlatmstc  36457  pnonsingN  37071  lhpocnle  37154  dochnoncon  38529