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Theorem posasymb 18366
Description: A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
posi.b 𝐵 = (Base‘𝐾)
posi.l = (le‘𝐾)
Assertion
Ref Expression
posasymb ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Proof of Theorem posasymb
StepHypRef Expression
1 simp1 1136 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Poset)
2 simp2 1137 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1138 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 posi.b . . . . 5 𝐵 = (Base‘𝐾)
5 posi.l . . . . 5 = (le‘𝐾)
64, 5posi 18364 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑌𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑌) → 𝑋 𝑌)))
71, 2, 3, 3, 6syl13anc 1373 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑌) → 𝑋 𝑌)))
87simp2d 1143 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
94, 5posref 18365 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
10 breq2 5146 . . . . 5 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
119, 10syl5ibcom 245 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑌𝑋 𝑌))
12 breq1 5145 . . . . 5 (𝑋 = 𝑌 → (𝑋 𝑋𝑌 𝑋))
139, 12syl5ibcom 245 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑌𝑌 𝑋))
1411, 13jcad 512 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑌 → (𝑋 𝑌𝑌 𝑋)))
15143adant3 1132 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 → (𝑋 𝑌𝑌 𝑋)))
168, 15impbid 212 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  Basecbs 17248  lecple 17305  Posetcpo 18354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-proset 18341  df-poset 18360
This theorem is referenced by:  odupos  18374  pltnle  18384  pltval3  18385  lublecllem  18406  poslubmo  18457  posglbmo  18458  latasymb  18488  latleeqj1  18497  latleeqm1  18513  posrasymb  32956  mgcf1olem1  32992  mgcf1olem2  32993  archirngz  33197  archiabllem1a  33199  ople0  39189  op1le  39194  atlle0  39307
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