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Theorem opexmid 35009
Description: Law of excluded middle for orthoposets. (chjo 28708 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 2770 . . . 4 (le‘𝐾) = (le‘𝐾)
3 opexmid.o . . . 4 = (oc‘𝐾)
4 opexmid.j . . . 4 = (join‘𝐾)
5 eqid 2770 . . . 4 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2770 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7 opexmid.u . . . 4 1 = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 34984 . . 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 1136 1 ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  occoc 16156  joincjn 17151  meetcmee 17152  0.cp0 17244  1.cp1 17245  OPcops 34974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-dm 5259  df-iota 5994  df-fv 6039  df-ov 6795  df-oposet 34978
This theorem is referenced by:  dih1  37089
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