MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  posasymb Structured version   Visualization version   GIF version

Theorem posasymb 18377
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 1135 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Poset)
2 simp2 1136 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1137 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 posi.b . . . . 5 𝐵 = (Base‘𝐾)
5 posi.l . . . . 5 = (le‘𝐾)
64, 5posi 18375 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑌𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑌) → 𝑋 𝑌)))
71, 2, 3, 3, 6syl13anc 1371 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑌) → 𝑋 𝑌)))
87simp2d 1142 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
94, 5posref 18376 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
10 breq2 5152 . . . . 5 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
119, 10syl5ibcom 245 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑌𝑋 𝑌))
12 breq1 5151 . . . . 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 1131 . 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 1537  wcel 2106   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-proset 18352  df-poset 18371
This theorem is referenced by:  odupos  18386  pltnle  18396  pltval3  18397  lublecllem  18418  poslubmo  18469  posglbmo  18470  latasymb  18500  latleeqj1  18509  latleeqm1  18525  posrasymb  32940  mgcf1olem1  32976  mgcf1olem2  32977  archirngz  33179  archiabllem1a  33181  ople0  39169  op1le  39174  atlle0  39287
  Copyright terms: Public domain W3C validator