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Theorem ispos2 18361
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 1097 . . . . . . 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 3093 . . . . . 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 3111 . . . . . 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 3128 . . . 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 3138 . . . . 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 3093 . . . . . 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 18360 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
1613, 14isprs 18342 . . . 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 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304   Proset cproset 18338  Posetcpo 18353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-proset 18340  df-poset 18359
This theorem is referenced by:  posprs  18362  postcposALT  49172  postc  49173
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