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Theorem ispos2 18373
Description: A poset is an antisymmetric proset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses
Ref Expression
ispos2.b 𝐵 = (Base‘𝐾)
ispos2.l = (le‘𝐾)
Assertion
Ref Expression
ispos2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝐾,𝑦   𝑥,𝐵,𝑦   𝑥, ,𝑦

Proof of Theorem ispos2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 3anan32 1096 . . . . . . 7 ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
21ralbii 3091 . . . . . 6 (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑧𝐵 ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
3 r19.26 3109 . . . . . 6 (∀𝑧𝐵 ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
42, 3bitri 275 . . . . 5 (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
542ralbii 3126 . . . 4 (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
6 r19.26-2 3136 . . . . 5 (∀𝑥𝐵𝑦𝐵 (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
7 rr19.3v 3667 . . . . . . 7 (∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ∀𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
87ralbii 3091 . . . . . 6 (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
98anbi2i 623 . . . . 5 ((∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
106, 9bitri 275 . . . 4 (∀𝑥𝐵𝑦𝐵 (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
115, 10bitri 275 . . 3 (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
1211anbi2i 623 . 2 ((𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))) ↔ (𝐾 ∈ V ∧ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))))
13 ispos2.b . . 3 𝐵 = (Base‘𝐾)
14 ispos2.l . . 3 = (le‘𝐾)
1513, 14ispos 18372 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
1613, 14isprs 18354 . . . 4 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
1716anbi1i 624 . . 3 ((𝐾 ∈ Proset ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ ((𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
18 anass 468 . . 3 (((𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ (𝐾 ∈ V ∧ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))))
1917, 18bitri 275 . 2 ((𝐾 ∈ Proset ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)) ↔ (𝐾 ∈ V ∧ (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))))
2012, 15, 193bitr4i 303 1 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀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  wral 3059  Vcvv 3478   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305   Proset cproset 18350  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:  posprs  18374  postcposALT  48884  postc  48885
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