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Theorem opexmid 34993
Description: Law of excluded middle for orthoposets. (chjo 28679 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 ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 1 )

Proof of Theorem opexmid
StepHypRef Expression
1 opexmid.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2756 . . . 4 (le‘𝐾) = (le‘𝐾)
3 opexmid.o . . . 4 = (oc‘𝐾)
4 opexmid.j . . . 4 = (join‘𝐾)
5 eqid 2756 . . . 4 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2756 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7 opexmid.u . . . 4 1 = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 34968 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑋𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( 𝑋)) = (0.‘𝐾)))
983anidm23 1530 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( 𝑋)(le‘𝐾)( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋(meet‘𝐾)( 𝑋)) = (0.‘𝐾)))
109simp2d 1138 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1628  wcel 2135   class class class wbr 4800  cfv 6045  (class class class)co 6809  Basecbs 16055  lecple 16146  occoc 16147  joincjn 17141  meetcmee 17142  0.cp0 17234  1.cp1 17235  OPcops 34958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-nul 4937
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-dm 5272  df-iota 6008  df-fv 6053  df-ov 6812  df-oposet 34962
This theorem is referenced by:  dih1  37073
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