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Theorem pospo 17180
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 17170 . . . 4 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → ¬ 𝑥 < 𝑥)
3 pospo.b . . . . 5 𝐵 = (Base‘𝐾)
43, 1plttr 17177 . . . 4 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 5178 . . 3 (𝐾 ∈ Poset → < Po 𝐵)
6 relres 5567 . . . . 5 Rel ( I ↾ 𝐵)
76a1i 11 . . . 4 (𝐾 ∈ Poset → Rel ( I ↾ 𝐵))
8 opabresid 5596 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} = ( I ↾ 𝐵)
98eleq2i 2841 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵))
10 opabid 5115 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} ↔ (𝑥𝐵𝑦 = 𝑥))
119, 10bitr3i 266 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) ↔ (𝑥𝐵𝑦 = 𝑥))
12 pospo.l . . . . . . . 8 = (le‘𝐾)
133, 12posref 17158 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → 𝑥 𝑥)
14 df-br 4785 . . . . . . . 8 (𝑥 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ )
15 breq2 4788 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 𝑦𝑥 𝑥))
1614, 15syl5bbr 274 . . . . . . 7 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ 𝑥 𝑥))
1713, 16syl5ibrcom 237 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ ∈ ))
1817expimpd 441 . . . . 5 (𝐾 ∈ Poset → ((𝑥𝐵𝑦 = 𝑥) → ⟨𝑥, 𝑦⟩ ∈ ))
1911, 18syl5bi 232 . . . 4 (𝐾 ∈ Poset → (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ))
207, 19relssdv 5352 . . 3 (𝐾 ∈ Poset → ( I ↾ 𝐵) ⊆ )
215, 20jca 495 . 2 (𝐾 ∈ Poset → ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ))
22 elex 3361 . . . . 5 (𝐾𝑉𝐾 ∈ V)
2322adantr 466 . . . 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 2096 . . . . . 6 𝑥 = 𝑥
27 simpr 471 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥𝐵)
28 resieq 5548 . . . . . . 7 ((𝑥𝐵𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 = 𝑥))
2927, 27, 28syl2anc 565 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 = 𝑥))
3026, 29mpbiri 248 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥( I ↾ 𝐵)𝑥)
31 simplrr 755 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → ( I ↾ 𝐵) ⊆ )
3231ssbrd 4827 . . . . 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 17171 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
35343adant1 1123 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
363, 12, 1pleval2i 17171 . . . . . . 7 ((𝑦𝐵𝑥𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
3736ancoms 455 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
38373adant1 1123 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
39 simprl 746 . . . . . . . 8 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → < Po 𝐵)
40 po2nr 5183 . . . . . . . . 9 (( < Po 𝐵 ∧ (𝑥𝐵𝑦𝐵)) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
41403impb 1106 . . . . . . . 8 (( < Po 𝐵𝑥𝐵𝑦𝐵) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
4239, 41syl3an1 1165 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
4342pm2.21d 119 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
44 simpl 468 . . . . . . 7 ((𝑥 = 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦)
4544a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 = 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
46 simpr 471 . . . . . . . 8 ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑦 = 𝑥)
4746eqcomd 2776 . . . . . . 7 ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
4847a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦))
49 simpl 468 . . . . . . 7 ((𝑥 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
5049a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦))
5143, 45, 48, 50ccased 1023 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (((𝑥 < 𝑦𝑥 = 𝑦) ∧ (𝑦 < 𝑥𝑦 = 𝑥)) → 𝑥 = 𝑦))
5235, 38, 51syl2and 587 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
53 simpr1 1232 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
54 simpr2 1234 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
5553, 54, 34syl2anc 565 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
56 simpr3 1236 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑧𝐵)
573, 12, 1pleval2i 17171 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧 → (𝑦 < 𝑧𝑦 = 𝑧)))
5854, 56, 57syl2anc 565 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑦 𝑧 → (𝑦 < 𝑧𝑦 = 𝑧)))
59 potr 5182 . . . . . . . 8 (( < Po 𝐵 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
6039, 59sylan 561 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
61 simpll 742 . . . . . . . 8 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝐾𝑉)
6212, 1pltle 17168 . . . . . . . 8 ((𝐾𝑉𝑥𝐵𝑧𝐵) → (𝑥 < 𝑧𝑥 𝑧))
6361, 53, 56, 62syl3anc 1475 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 < 𝑧𝑥 𝑧))
6460, 63syld 47 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 𝑧))
65 breq1 4787 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 < 𝑧𝑦 < 𝑧))
6665biimpar 463 . . . . . . 7 ((𝑥 = 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧)
6766, 63syl5 34 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 = 𝑦𝑦 < 𝑧) → 𝑥 𝑧))
68 breq2 4788 . . . . . . . 8 (𝑦 = 𝑧 → (𝑥 < 𝑦𝑥 < 𝑧))
6968biimpac 464 . . . . . . 7 ((𝑥 < 𝑦𝑦 = 𝑧) → 𝑥 < 𝑧)
7069, 63syl5 34 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 = 𝑧) → 𝑥 𝑧))
7153, 33syldan 571 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥 𝑥)
72 eqtr 2789 . . . . . . . 8 ((𝑥 = 𝑦𝑦 = 𝑧) → 𝑥 = 𝑧)
7372breq2d 4796 . . . . . . 7 ((𝑥 = 𝑦𝑦 = 𝑧) → (𝑥 𝑥𝑥 𝑧))
7471, 73syl5ibcom 235 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 = 𝑦𝑦 = 𝑧) → 𝑥 𝑧))
7564, 67, 70, 74ccased 1023 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (((𝑥 < 𝑦𝑥 = 𝑦) ∧ (𝑦 < 𝑧𝑦 = 𝑧)) → 𝑥 𝑧))
7655, 58, 75syl2and 587 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
7723, 24, 25, 33, 52, 76isposd 17162 . . 3 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → 𝐾 ∈ Poset)
7877ex 397 . 2 (𝐾𝑉 → (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) → 𝐾 ∈ Poset))
7921, 78impbid2 216 1 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  Vcvv 3349  wss 3721  cop 4320   class class class wbr 4784  {copab 4844   I cid 5156   Po wpo 5168  cres 5251  Rel wrel 5254  cfv 6031  Basecbs 16063  lecple 16155  Posetcpo 17147  ltcplt 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-po 5170  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-preset 17135  df-poset 17153  df-plt 17165
This theorem is referenced by:  tosso  17243
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