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Theorem pospo 17412
 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 17402 . . . 4 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → ¬ 𝑥 < 𝑥)
3 pospo.b . . . . 5 𝐵 = (Base‘𝐾)
43, 1plttr 17409 . . . 4 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
52, 4ispod 5370 . . 3 (𝐾 ∈ Poset → < Po 𝐵)
6 relres 5763 . . . . 5 Rel ( I ↾ 𝐵)
76a1i 11 . . . 4 (𝐾 ∈ Poset → Rel ( I ↾ 𝐵))
8 opabresid 5797 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} = ( I ↾ 𝐵)
98eleq2i 2874 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵))
10 opabid 5303 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝑥)} ↔ (𝑥𝐵𝑦 = 𝑥))
119, 10bitr3i 278 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) ↔ (𝑥𝐵𝑦 = 𝑥))
12 pospo.l . . . . . . . 8 = (le‘𝐾)
133, 12posref 17390 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → 𝑥 𝑥)
14 df-br 4963 . . . . . . . 8 (𝑥 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ )
15 breq2 4966 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 𝑦𝑥 𝑥))
1614, 15syl5bbr 286 . . . . . . 7 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ 𝑥 𝑥))
1713, 16syl5ibrcom 248 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑥𝐵) → (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ ∈ ))
1817expimpd 454 . . . . 5 (𝐾 ∈ Poset → ((𝑥𝐵𝑦 = 𝑥) → ⟨𝑥, 𝑦⟩ ∈ ))
1911, 18syl5bi 243 . . . 4 (𝐾 ∈ Poset → (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ))
207, 19relssdv 5547 . . 3 (𝐾 ∈ Poset → ( I ↾ 𝐵) ⊆ )
215, 20jca 512 . 2 (𝐾 ∈ Poset → ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ))
22 elex 3455 . . . . 5 (𝐾𝑉𝐾 ∈ V)
2322adantr 481 . . . 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 1996 . . . . . 6 𝑥 = 𝑥
27 simpr 485 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥𝐵)
28 resieq 5745 . . . . . . 7 ((𝑥𝐵𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 = 𝑥))
2927, 27, 28syl2anc 584 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → (𝑥( I ↾ 𝐵)𝑥𝑥 = 𝑥))
3026, 29mpbiri 259 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → 𝑥( I ↾ 𝐵)𝑥)
31 simplrr 774 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵) → ( I ↾ 𝐵) ⊆ )
3231ssbrd 5005 . . . . 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 17403 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
35343adant1 1123 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
363, 12, 1pleval2i 17403 . . . . . . 7 ((𝑦𝐵𝑥𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
3736ancoms 459 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
38373adant1 1123 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 → (𝑦 < 𝑥𝑦 = 𝑥)))
39 simprl 767 . . . . . . . 8 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → < Po 𝐵)
40 po2nr 5375 . . . . . . . . 9 (( < Po 𝐵 ∧ (𝑥𝐵𝑦𝐵)) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
41403impb 1108 . . . . . . . 8 (( < Po 𝐵𝑥𝐵𝑦𝐵) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
4239, 41syl3an1 1156 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ¬ (𝑥 < 𝑦𝑦 < 𝑥))
4342pm2.21d 121 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
44 simpl 483 . . . . . . 7 ((𝑥 = 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦)
4544a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 = 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
46 simpr 485 . . . . . . . 8 ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑦 = 𝑥)
4746equcomd 2003 . . . . . . 7 ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
4847a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 < 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦))
49 simpl 483 . . . . . . 7 ((𝑥 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
5049a1i 11 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦))
5143, 45, 48, 50ccased 1031 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → (((𝑥 < 𝑦𝑥 = 𝑦) ∧ (𝑦 < 𝑥𝑦 = 𝑥)) → 𝑥 = 𝑦))
5235, 38, 51syl2and 607 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ 𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
53 simpr1 1187 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
54 simpr2 1188 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
5553, 54, 34syl2anc 584 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 𝑦 → (𝑥 < 𝑦𝑥 = 𝑦)))
56 simpr3 1189 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑧𝐵)
573, 12, 1pleval2i 17403 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧 → (𝑦 < 𝑧𝑦 = 𝑧)))
5854, 56, 57syl2anc 584 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑦 𝑧 → (𝑦 < 𝑧𝑦 = 𝑧)))
59 potr 5374 . . . . . . . 8 (( < Po 𝐵 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
6039, 59sylan 580 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧))
61 simpll 763 . . . . . . . 8 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝐾𝑉)
6212, 1pltle 17400 . . . . . . . 8 ((𝐾𝑉𝑥𝐵𝑧𝐵) → (𝑥 < 𝑧𝑥 𝑧))
6361, 53, 56, 62syl3anc 1364 . . . . . . 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 4965 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 < 𝑧𝑦 < 𝑧))
6665biimpar 478 . . . . . . 7 ((𝑥 = 𝑦𝑦 < 𝑧) → 𝑥 < 𝑧)
6766, 63syl5 34 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 = 𝑦𝑦 < 𝑧) → 𝑥 𝑧))
68 breq2 4966 . . . . . . . 8 (𝑦 = 𝑧 → (𝑥 < 𝑦𝑥 < 𝑧))
6968biimpac 479 . . . . . . 7 ((𝑥 < 𝑦𝑦 = 𝑧) → 𝑥 < 𝑧)
7069, 63syl5 34 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 < 𝑦𝑦 = 𝑧) → 𝑥 𝑧))
7153, 33syldan 591 . . . . . . 7 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥 𝑥)
72 eqtr 2816 . . . . . . . 8 ((𝑥 = 𝑦𝑦 = 𝑧) → 𝑥 = 𝑧)
7372breq2d 4974 . . . . . . 7 ((𝑥 = 𝑦𝑦 = 𝑧) → (𝑥 𝑥𝑥 𝑧))
7471, 73syl5ibcom 246 . . . . . 6 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 = 𝑦𝑦 = 𝑧) → 𝑥 𝑧))
7564, 67, 70, 74ccased 1031 . . . . 5 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (((𝑥 < 𝑦𝑥 = 𝑦) ∧ (𝑦 < 𝑧𝑦 = 𝑧)) → 𝑥 𝑧))
7655, 58, 75syl2and 607 . . . 4 (((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
7723, 24, 25, 33, 52, 76isposd 17394 . . 3 ((𝐾𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )) → 𝐾 ∈ Poset)
7877ex 413 . 2 (𝐾𝑉 → (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) → 𝐾 ∈ Poset))
7921, 78impbid2 227 1 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ⊆ wss 3859  ⟨cop 4478   class class class wbr 4962  {copab 5024   I cid 5347   Po wpo 5360   ↾ cres 5445  Rel wrel 5448  ‘cfv 6225  Basecbs 16312  lecple 16401  Posetcpo 17379  ltcplt 17380 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-res 5455  df-iota 6189  df-fun 6227  df-fv 6233  df-proset 17367  df-poset 17385  df-plt 17397 This theorem is referenced by:  tosso  17475
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