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Theorem oplecon1b 36217
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 29205 analog.) (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐵 = (Base‘𝐾)
opcon3.l = (le‘𝐾)
opcon3.o = (oc‘𝐾)
Assertion
Ref Expression
oplecon1b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))

Proof of Theorem oplecon1b
StepHypRef Expression
1 opcon3.b . . . . 5 𝐵 = (Base‘𝐾)
2 opcon3.o . . . . 5 = (oc‘𝐾)
31, 2opoccl 36210 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
433adant3 1124 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
5 opcon3.l . . . 4 = (le‘𝐾)
61, 5, 2oplecon3b 36216 . . 3 ((𝐾 ∈ OP ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
74, 6syld3an2 1403 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) ( ‘( 𝑋))))
81, 2opococ 36211 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
983adant3 1124 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
109breq2d 5069 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( ‘( 𝑋)) ↔ ( 𝑌) 𝑋))
117, 10bitrd 280 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  occoc 16561  OPcops 36188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148  df-oposet 36192
This theorem is referenced by:  opoc1  36218  oldmm1  36233
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