Theorem ispos 18520

Description: The predicate "is a poset". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)

Hypotheses

Ref	Expression
ispos.b	⊢ 𝐵 = (Base‘𝐾)
ispos.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
ispos	⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥, ≤ ,𝑦,𝑧

Allowed substitution hints:   𝐾(𝑥, 𝑦, 𝑧)

Proof of Theorem ispos

Dummy variables 𝑝 𝑏 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6911	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2		ispos.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2816	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4	3	eqeq2d 2774	. . . . 5 ⊢ (𝑝 = 𝐾 → (𝑏 = (Base‘𝑝) ↔ 𝑏 = 𝐵))
5		fveq2 6911	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
6		ispos.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
7	5, 6	eqtr4di 2816	. . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
8	7	eqeq2d 2774	. . . . 5 ⊢ (𝑝 = 𝐾 → (𝑟 = (le‘𝑝) ↔ 𝑟 = ≤ ))
9	4, 8	3anbi12d 1466	. . . 4 ⊢ (𝑝 = 𝐾 → ((𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ (𝑏 = 𝐵 ∧ 𝑟 = ≤ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
10	9	2exbidv 1959	. . 3 ⊢ (𝑝 = 𝐾 → (∃𝑏∃𝑟(𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ ∃𝑏∃𝑟(𝑏 = 𝐵 ∧ 𝑟 = ≤ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
11		df-poset 18519	. . 3 ⊢ Poset = {𝑝 ∣ ∃𝑏∃𝑟(𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}
12	10, 11	elab4g 3640	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∃𝑏∃𝑟(𝑏 = 𝐵 ∧ 𝑟 = ≤ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
13	2	fvexi 6925	. . . 4 ⊢ 𝐵 ∈ V
14	6	fvexi 6925	. . . 4 ⊢ ≤ ∈ V
15		raleq 3320	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
16	15	raleqbi1dv 3333	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
17	16	raleqbi1dv 3333	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
18		breq 5108	. . . . . . 7 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑥 ↔ 𝑥 ≤ 𝑥))
19		breq 5108	. . . . . . . . 9 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑦 ↔ 𝑥 ≤ 𝑦))
20		breq 5108	. . . . . . . . 9 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑥 ↔ 𝑦 ≤ 𝑥))
21	19, 20	anbi12d 644	. . . . . . . 8 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
22	21	imbi1d 344	. . . . . . 7 ⊢ (𝑟 = ≤ → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ↔ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)))
23		breq 5108	. . . . . . . . 9 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑧 ↔ 𝑦 ≤ 𝑧))
24	19, 23	anbi12d 644	. . . . . . . 8 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧)))
25		breq 5108	. . . . . . . 8 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑧 ↔ 𝑥 ≤ 𝑧))
26	24, 25	imbi12d 347	. . . . . . 7 ⊢ (𝑟 = ≤ → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
27	18, 22, 26	3anbi123d 1465	. . . . . 6 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
28	27	ralbidv 3189	. . . . 5 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
29	28	2ralbidv 3230	. . . 4 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
30	13, 14, 17, 29	ceqsex2v 3505	. . 3 ⊢ (∃𝑏∃𝑟(𝑏 = 𝐵 ∧ 𝑟 = ≤ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
31	30	anbi2i 635	. 2 ⊢ ((𝐾 ∈ V ∧ ∃𝑏∃𝑟(𝑏 = 𝐵 ∧ 𝑟 = ≤ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))) ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
32	12, 31	bitri 278	1 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   class class class wbr 5106  ‘cfv 6561  Basecbs 17419  lecple 17467  Posetcpo 18513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-poset 18519

This theorem is used by:  ispos2  18521  posi  18523  0pos  18527  isposd  18528  isposi  18529  pospropd  18531  resspos  18635  zsoring  28828