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Theorem ispos2 17546
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 1089 . . . . . . 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 3162 . . . . . 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 3167 . . . . . 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 276 . . . . 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 3163 . . . 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 3168 . . . . 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 3658 . . . . . . 7 (∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ∀𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
87ralbii 3162 . . . . . 6 (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
98anbi2i 622 . . . . 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 276 . . . 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 276 . . 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 622 . 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 17545 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
1613, 14isprs 17528 . . . 4 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
1716anbi1i 623 . . 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 469 . . 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 276 . 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 304 1 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560   Proset cproset 17524  Posetcpo 17538
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-iota 6307  df-fv 6356  df-proset 17526  df-poset 17544
This theorem is referenced by:  posprs  17547
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