Theorem pospo 18063

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	⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

Proof of Theorem pospo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pospo.s	. . . . 5 ⊢ < = (lt‘𝐾)
2	1	pltirr 18053	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 < 𝑥)
3		pospo.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4	3, 1	plttr 18060	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
5	2, 4	ispod 5512	. . 3 ⊢ (𝐾 ∈ Poset → < Po 𝐵)
6		relres 5920	. . . . 5 ⊢ Rel ( I ↾ 𝐵)
7	6	a1i 11	. . . 4 ⊢ (𝐾 ∈ Poset → Rel ( I ↾ 𝐵))
8		opabresid 5957	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)}
9	8	eqcomi 2747	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐵)
10	9	eleq2i 2830	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵))
11		opabidw 5437	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥))
12	10, 11	bitr3i 276	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥))
13		pospo.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
14	3, 13	posref 18036	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
15		df-br 5075	. . . . . . . 8 ⊢ (𝑥 ≤ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≤ )
16		breq2 5078	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑥))
17	15, 16	bitr3id 285	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ≤ ↔ 𝑥 ≤ 𝑥))
18	14, 17	syl5ibrcom 246	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
19	18	expimpd 454	. . . . 5 ⊢ (𝐾 ∈ Poset → ((𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥) → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
20	12, 19	syl5bi 241	. . . 4 ⊢ (𝐾 ∈ Poset → (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
21	7, 20	relssdv 5698	. . 3 ⊢ (𝐾 ∈ Poset → ( I ↾ 𝐵) ⊆ ≤ )
22	5, 21	jca 512	. 2 ⊢ (𝐾 ∈ Poset → ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ))
23		simpl 483	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → 𝐾 ∈ 𝑉)
24	3	a1i 11	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → 𝐵 = (Base‘𝐾))
25	13	a1i 11	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → ≤ = (le‘𝐾))
26		equid 2015	. . . . . 6 ⊢ 𝑥 = 𝑥
27		simpr 485	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
28		resieq 5902	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥( I ↾ 𝐵)𝑥 ↔ 𝑥 = 𝑥))
29	27, 27, 28	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → (𝑥( I ↾ 𝐵)𝑥 ↔ 𝑥 = 𝑥))
30	26, 29	mpbiri 257	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → 𝑥( I ↾ 𝐵)𝑥)
31		simplrr 775	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → ( I ↾ 𝐵) ⊆ ≤ )
32	31	ssbrd 5117	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → (𝑥( I ↾ 𝐵)𝑥 → 𝑥 ≤ 𝑥))
33	30, 32	mpd 15	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
34	3, 13, 1	pleval2i 18054	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
35	34	3adant1 1129	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
36	3, 13, 1	pleval2i 18054	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
37	36	ancoms 459	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
38	37	3adant1 1129	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
39		simprl 768	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → < Po 𝐵)
40		po2nr 5517	. . . . . . . . 9 ⊢ (( < Po 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
41	40	3impb 1114	. . . . . . . 8 ⊢ (( < Po 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
42	39, 41	syl3an1 1162	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
43	42	pm2.21d 121	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑥) → 𝑥 = 𝑦))
44		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 < 𝑥) → 𝑥 = 𝑦)
45	44	a1i 11	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 = 𝑦 ∧ 𝑦 < 𝑥) → 𝑥 = 𝑦))
46		simpr 485	. . . . . . . 8 ⊢ ((𝑥 < 𝑦 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
47	46	equcomd 2022	. . . . . . 7 ⊢ ((𝑥 < 𝑦 ∧ 𝑦 = 𝑥) → 𝑥 = 𝑦)
48	47	a1i 11	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 < 𝑦 ∧ 𝑦 = 𝑥) → 𝑥 = 𝑦))
49		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝑥) → 𝑥 = 𝑦)
50	49	a1i 11	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 = 𝑦 ∧ 𝑦 = 𝑥) → 𝑥 = 𝑦))
51	43, 45, 48, 50	ccased 1036	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∧ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → 𝑥 = 𝑦))
52	35, 38, 51	syl2and 608	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
53		simpr1 1193	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
54		simpr2 1194	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
55	53, 54, 34	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
56		simpr3 1195	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
57	3, 13, 1	pleval2i 18054	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ≤ 𝑧 → (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)))
58	54, 56, 57	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ≤ 𝑧 → (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)))
59		potr 5516	. . . . . . . 8 ⊢ (( < Po 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
60	39, 59	sylan 580	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
61		simpll 764	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
62	13, 1	pltle 18051	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 < 𝑧 → 𝑥 ≤ 𝑧))
63	61, 53, 56, 62	syl3anc 1370	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 < 𝑧 → 𝑥 ≤ 𝑧))
64	60, 63	syld 47	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 ≤ 𝑧))
65		breq1 5077	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 < 𝑧 ↔ 𝑦 < 𝑧))
66	65	biimpar 478	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)
67	66, 63	syl5 34	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 = 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 ≤ 𝑧))
68		breq2 5078	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑥 < 𝑦 ↔ 𝑥 < 𝑧))
69	68	biimpac 479	. . . . . . 7 ⊢ ((𝑥 < 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 < 𝑧)
70	69, 63	syl5 34	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 ≤ 𝑧))
71	53, 33	syldan 591	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ≤ 𝑥)
72		eqtr 2761	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑧)
73	72	breq2d 5086	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝑧) → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑧))
74	71, 73	syl5ibcom 244	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 = 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 ≤ 𝑧))
75	64, 67, 70, 74	ccased 1036	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∧ (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)) → 𝑥 ≤ 𝑧))
76	55, 58, 75	syl2and 608	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
77	23, 24, 25, 33, 52, 76	isposd 18041	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → 𝐾 ∈ Poset)
78	77	ex 413	. 2 ⊢ (𝐾 ∈ 𝑉 → (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) → 𝐾 ∈ Poset))
79	22, 78	impbid2 225	1 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074  {copab 5136   I cid 5488   Po wpo 5501   ↾ cres 5591  Rel wrel 5594  ‘cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  ltcplt 18026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-proset 18013  df-poset 18031  df-plt 18048

This theorem is referenced by:  tosso  18137