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Theorem ispos 18266
Description: The predicate "is a poset". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
Hypotheses
Ref Expression
ispos.b 𝐡 = (Baseβ€˜πΎ)
ispos.l ≀ = (leβ€˜πΎ)
Assertion
Ref Expression
ispos (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐡   π‘₯, ≀ ,𝑦,𝑧
Allowed substitution hints:   𝐾(π‘₯,𝑦,𝑧)

Proof of Theorem ispos
Dummy variables 𝑝 𝑏 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . . . 7 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΎ))
2 ispos.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
31, 2eqtr4di 2790 . . . . . 6 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = 𝐡)
43eqeq2d 2743 . . . . 5 (𝑝 = 𝐾 β†’ (𝑏 = (Baseβ€˜π‘) ↔ 𝑏 = 𝐡))
5 fveq2 6891 . . . . . . 7 (𝑝 = 𝐾 β†’ (leβ€˜π‘) = (leβ€˜πΎ))
6 ispos.l . . . . . . 7 ≀ = (leβ€˜πΎ)
75, 6eqtr4di 2790 . . . . . 6 (𝑝 = 𝐾 β†’ (leβ€˜π‘) = ≀ )
87eqeq2d 2743 . . . . 5 (𝑝 = 𝐾 β†’ (π‘Ÿ = (leβ€˜π‘) ↔ π‘Ÿ = ≀ ))
94, 83anbi12d 1437 . . . 4 (𝑝 = 𝐾 β†’ ((𝑏 = (Baseβ€˜π‘) ∧ π‘Ÿ = (leβ€˜π‘) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§))) ↔ (𝑏 = 𝐡 ∧ π‘Ÿ = ≀ ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)))))
1092exbidv 1927 . . 3 (𝑝 = 𝐾 β†’ (βˆƒπ‘βˆƒπ‘Ÿ(𝑏 = (Baseβ€˜π‘) ∧ π‘Ÿ = (leβ€˜π‘) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§))) ↔ βˆƒπ‘βˆƒπ‘Ÿ(𝑏 = 𝐡 ∧ π‘Ÿ = ≀ ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)))))
11 df-poset 18265 . . 3 Poset = {𝑝 ∣ βˆƒπ‘βˆƒπ‘Ÿ(𝑏 = (Baseβ€˜π‘) ∧ π‘Ÿ = (leβ€˜π‘) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)))}
1210, 11elab4g 3673 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ βˆƒπ‘βˆƒπ‘Ÿ(𝑏 = 𝐡 ∧ π‘Ÿ = ≀ ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)))))
132fvexi 6905 . . . 4 𝐡 ∈ V
146fvexi 6905 . . . 4 ≀ ∈ V
15 raleq 3322 . . . . . 6 (𝑏 = 𝐡 β†’ (βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)) ↔ βˆ€π‘§ ∈ 𝐡 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§))))
1615raleqbi1dv 3333 . . . . 5 (𝑏 = 𝐡 β†’ (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)) ↔ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§))))
1716raleqbi1dv 3333 . . . 4 (𝑏 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§))))
18 breq 5150 . . . . . . 7 (π‘Ÿ = ≀ β†’ (π‘₯π‘Ÿπ‘₯ ↔ π‘₯ ≀ π‘₯))
19 breq 5150 . . . . . . . . 9 (π‘Ÿ = ≀ β†’ (π‘₯π‘Ÿπ‘¦ ↔ π‘₯ ≀ 𝑦))
20 breq 5150 . . . . . . . . 9 (π‘Ÿ = ≀ β†’ (π‘¦π‘Ÿπ‘₯ ↔ 𝑦 ≀ π‘₯))
2119, 20anbi12d 631 . . . . . . . 8 (π‘Ÿ = ≀ β†’ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) ↔ (π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯)))
2221imbi1d 341 . . . . . . 7 (π‘Ÿ = ≀ β†’ (((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ↔ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦)))
23 breq 5150 . . . . . . . . 9 (π‘Ÿ = ≀ β†’ (π‘¦π‘Ÿπ‘§ ↔ 𝑦 ≀ 𝑧))
2419, 23anbi12d 631 . . . . . . . 8 (π‘Ÿ = ≀ β†’ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) ↔ (π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧)))
25 breq 5150 . . . . . . . 8 (π‘Ÿ = ≀ β†’ (π‘₯π‘Ÿπ‘§ ↔ π‘₯ ≀ 𝑧))
2624, 25imbi12d 344 . . . . . . 7 (π‘Ÿ = ≀ β†’ (((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§) ↔ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)))
2718, 22, 263anbi123d 1436 . . . . . 6 (π‘Ÿ = ≀ β†’ ((π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)) ↔ (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
2827ralbidv 3177 . . . . 5 (π‘Ÿ = ≀ β†’ (βˆ€π‘§ ∈ 𝐡 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)) ↔ βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
29282ralbidv 3218 . . . 4 (π‘Ÿ = ≀ β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
3013, 14, 17, 29ceqsex2v 3530 . . 3 (βˆƒπ‘βˆƒπ‘Ÿ(𝑏 = 𝐡 ∧ π‘Ÿ = ≀ ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§))) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)))
3130anbi2i 623 . 2 ((𝐾 ∈ V ∧ βˆƒπ‘βˆƒπ‘Ÿ(𝑏 = 𝐡 ∧ π‘Ÿ = ≀ ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 (π‘₯π‘Ÿπ‘₯ ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯π‘Ÿπ‘¦ ∧ π‘¦π‘Ÿπ‘§) β†’ π‘₯π‘Ÿπ‘§)))) ↔ (𝐾 ∈ V ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
3212, 31bitri 274 1 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦) ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   class class class wbr 5148  β€˜cfv 6543  Basecbs 17143  lecple 17203  Posetcpo 18259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-poset 18265
This theorem is referenced by:  ispos2  18267  posi  18269  0pos  18273  0posOLD  18274  isposd  18275  isposi  18276  pospropd  18279  resspos  32131
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