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Theorem pospo 16796
Description: Write a poset structure in terms of the proper-class poset predicate (strict less than version). (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pospo.b 𝐵 = (Base‘𝐾)
pospo.l = (le‘𝐾)
pospo.s < = (lt‘𝐾)
Assertion
Ref Expression
pospo (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))

Proof of Theorem pospo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pospo.s . . . . 5 < = (lt‘𝐾)
21pltirr 16786 . . . 4 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → ¬ 𝑥 < 𝑥)
3 pospo.b . . . . 5 𝐵 = (Base‘𝐾)
43, 1plttr 16793 . . . 4 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 4967 . . 3 (𝐾 ∈ Poset → < Po 𝐵)
6 relres 5346 . . . . 5 Rel ( I ↾ 𝐵)
76a1i 11 . . . 4 (𝐾 ∈ Poset → Rel ( I ↾ 𝐵))
8 opabresid 5374 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} = ( I ↾ 𝐵)
98eleq2i 2680 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵))
10 opabid 4907 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} ↔ (𝑥𝐵𝑦 = 𝑥))
119, 10bitr3i 265 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) ↔ (𝑥𝐵𝑦 = 𝑥))
12 pospo.l . . . . . . . 8 = (le‘𝐾)
133, 12posref 16774 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → 𝑥 𝑥)
14 df-br 4584 . . . . . . . 8 (𝑥 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ )
15 breq2 4587 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 𝑦𝑥 𝑥))
1614, 15syl5bbr 273 . . . . . . 7 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ 𝑥 𝑥))
1713, 16syl5ibrcom 236 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ ∈ ))
1817expimpd 627 . . . . 5 (𝐾 ∈ Poset → ((𝑥𝐵𝑦 = 𝑥) → ⟨𝑥, 𝑦⟩ ∈ ))
1911, 18syl5bi 231 . . . 4 (𝐾 ∈ Poset → (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ))
207, 19relssdv 5135 . . 3 (𝐾 ∈ Poset → ( I ↾ 𝐵) ⊆ )
215, 20jca 553 . 2 (𝐾 ∈ Poset → ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ))
22 elex 3185 . . . . 5 (𝐾𝑉𝐾 ∈ V)
2322adantr 480 . . . 4 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → 𝐾 ∈ V)
243a1i 11 . . . 4 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → 𝐵 = (Base‘𝐾))
2512a1i 11 . . . 4 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → = (le‘𝐾))
26 equid 1926 . . . . . 6 𝑥 = 𝑥
27 simpr 476 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥𝐵)
28 resieq 5327 . . . . . . 7 ((𝑥𝐵𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 = 𝑥))
2927, 27, 28syl2anc 691 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 = 𝑥))
3026, 29mpbiri 247 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥( I ↾ 𝐵)𝑥)
31 simplrr 797 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → ( I ↾ 𝐵) ⊆ )
3231ssbrd 4626 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 𝑥))
3330, 32mpd 15 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥 𝑥)
343, 12, 1pleval2i 16787 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
35343adant1 1072 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
363, 12, 1pleval2i 16787 . . . . . . 7 ((𝑦𝐵𝑥𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
3736ancoms 468 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
38373adant1 1072 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
39 simprl 790 . . . . . . . 8 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → < Po 𝐵)
40 po2nr 4972 . . . . . . . . 9 (( < Po 𝐵 ∧ (𝑥𝐵𝑦𝐵)) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
41403impb 1252 . . . . . . . 8 (( < Po 𝐵𝑥𝐵𝑦𝐵) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
4239, 41syl3an1 1351 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
4342pm2.21d 117 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
44 simpl 472 . . . . . . 7 ((𝑥 = 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦)
4544a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 = 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
46 simpr 476 . . . . . . . 8 ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑦 = 𝑥)
4746eqcomd 2616 . . . . . . 7 ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
4847a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦))
49 simpl 472 . . . . . . 7 ((𝑥 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
5049a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦))
5143, 45, 48, 50ccased 985 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (((𝑥 < 𝑦𝑥 = 𝑦) ∧ (𝑦 < 𝑥𝑦 = 𝑥)) → 𝑥 = 𝑦))
5235, 38, 51syl2and 499 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
53 simpr1 1060 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
54 simpr2 1061 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
5553, 54, 34syl2anc 691 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
56 simpr3 1062 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑧𝐵)
573, 12, 1pleval2i 16787 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧 → (𝑦 < 𝑧𝑦 = 𝑧)))
5854, 56, 57syl2anc 691 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑦 𝑧 → (𝑦 < 𝑧𝑦 = 𝑧)))
59 potr 4971 . . . . . . . 8 (( < Po 𝐵 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
6039, 59sylan 487 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
61 simpll 786 . . . . . . . 8 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝐾𝑉)
6212, 1pltle 16784 . . . . . . . 8 ((𝐾𝑉𝑥𝐵𝑧𝐵) → (𝑥 < 𝑧𝑥 𝑧))
6361, 53, 56, 62syl3anc 1318 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 < 𝑧𝑥 𝑧))
6460, 63syld 46 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 𝑧))
65 breq1 4586 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 < 𝑧𝑦 < 𝑧))
6665biimpar 501 . . . . . . 7 ((𝑥 = 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧)
6766, 63syl5 33 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 = 𝑦𝑦 < 𝑧) → 𝑥 𝑧))
68 breq2 4587 . . . . . . . 8 (𝑦 = 𝑧 → (𝑥 < 𝑦𝑥 < 𝑧))
6968biimpac 502 . . . . . . 7 ((𝑥 < 𝑦𝑦 = 𝑧) → 𝑥 < 𝑧)
7069, 63syl5 33 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 = 𝑧) → 𝑥 𝑧))
7153, 33syldan 486 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥 𝑥)
72 eqtr 2629 . . . . . . . 8 ((𝑥 = 𝑦𝑦 = 𝑧) → 𝑥 = 𝑧)
7372breq2d 4595 . . . . . . 7 ((𝑥 = 𝑦𝑦 = 𝑧) → (𝑥 𝑥𝑥 𝑧))
7471, 73syl5ibcom 234 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 = 𝑦𝑦 = 𝑧) → 𝑥 𝑧))
7564, 67, 70, 74ccased 985 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (((𝑥 < 𝑦𝑥 = 𝑦) ∧ (𝑦 < 𝑧𝑦 = 𝑧)) → 𝑥 𝑧))
7655, 58, 75syl2and 499 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
7723, 24, 25, 33, 52, 76isposd 16778 . . 3 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → 𝐾 ∈ Poset)
7877ex 449 . 2 (𝐾𝑉 → (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) → 𝐾 ∈ Poset))
7921, 78impbid2 215 1 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540  cop 4131   class class class wbr 4583  {copab 4642   I cid 4948   Po wpo 4957  cres 5040  Rel wrel 5043  cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781
This theorem is referenced by:  tosso  16859
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