Theorem pospo 18497

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 18487	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 < 𝑥)
3		pospo.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4	3, 1	plttr 18494	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
5	2, 4	ispod 5568	. . 3 ⊢ (𝐾 ∈ Poset → < Po 𝐵)
6		relres 5996	. . . . 5 ⊢ Rel ( I ↾ 𝐵)
7	6	a1i 11	. . . 4 ⊢ (𝐾 ∈ Poset → Rel ( I ↾ 𝐵))
8		opabresid 6044	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)}
9	8	eqcomi 2770	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐵)
10	9	eleq2i 2853	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵))
11		opabidw 5498	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥))
12	10, 11	bitr3i 280	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥))
13		pospo.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
14	3, 13	posref 18472	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
15		df-br 5104	. . . . . . . 8 ⊢ (𝑥 ≤ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≤ )
16		breq2 5107	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑥))
17	15, 16	bitr3id 288	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ≤ ↔ 𝑥 ≤ 𝑥))
18	14, 17	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
19	18	expimpd 459	. . . . 5 ⊢ (𝐾 ∈ Poset → ((𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥) → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
20	12, 19	biimtrid 245	. . . 4 ⊢ (𝐾 ∈ Poset → (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
21	7, 20	relssdv 5764	. . 3 ⊢ (𝐾 ∈ Poset → ( I ↾ 𝐵) ⊆ ≤ )
22	5, 21	jca 521	. 2 ⊢ (𝐾 ∈ Poset → ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ))
23		simpl 488	. . . 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 2045	. . . . . 6 ⊢ 𝑥 = 𝑥
27		simpr 490	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
28		resieq 5981	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥( I ↾ 𝐵)𝑥 ↔ 𝑥 = 𝑥))
29	27, 27, 28	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → (𝑥( I ↾ 𝐵)𝑥 ↔ 𝑥 = 𝑥))
30	26, 29	mpbiri 261	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → 𝑥( I ↾ 𝐵)𝑥)
31		simplrr 790	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → ( I ↾ 𝐵) ⊆ ≤ )
32	31	ssbrd 5148	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → (𝑥( I ↾ 𝐵)𝑥 → 𝑥 ≤ 𝑥))
33	30, 32	mpd 16	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
34	3, 13, 1	pleval2i 18488	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
35	34	3adant1 1148	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
36	3, 13, 1	pleval2i 18488	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
37	36	ancoms 464	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
38	37	3adant1 1148	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
39		simprl 783	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → < Po 𝐵)
40		po2nr 5573	. . . . . . . . 9 ⊢ (( < Po 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
41	40	3impb 1132	. . . . . . . 8 ⊢ (( < Po 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
42	39, 41	syl3an1 1181	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
43	42	pm2.21d 122	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑥) → 𝑥 = 𝑦))
44		simpl 488	. . . . . . 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 490	. . . . . . . 8 ⊢ ((𝑥 < 𝑦 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
47	46	equcomd 2052	. . . . . . 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 488	. . . . . . 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 1054	. . . . 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 620	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
53		simpr1 1213	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
54		simpr2 1214	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
55	53, 54, 34	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
56		simpr3 1215	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
57	3, 13, 1	pleval2i 18488	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ≤ 𝑧 → (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)))
58	54, 56, 57	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ≤ 𝑧 → (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)))
59		potr 5572	. . . . . . . 8 ⊢ (( < Po 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
60	39, 59	sylan 592	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
61		simpll 779	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
62	13, 1	pltle 18485	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 < 𝑧 → 𝑥 ≤ 𝑧))
63	61, 53, 56, 62	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 < 𝑧 → 𝑥 ≤ 𝑧))
64	60, 63	syld 48	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 ≤ 𝑧))
65		breq1 5106	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 < 𝑧 ↔ 𝑦 < 𝑧))
66	65	biimpar 483	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)
67	66, 63	syl5 35	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 = 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 ≤ 𝑧))
68		breq2 5107	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑥 < 𝑦 ↔ 𝑥 < 𝑧))
69	68	biimpac 484	. . . . . . 7 ⊢ ((𝑥 < 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 < 𝑧)
70	69, 63	syl5 35	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 ≤ 𝑧))
71	53, 33	syldan 603	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ≤ 𝑥)
72		eqtr 2781	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑧)
73	72	breq2d 5115	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝑧) → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑧))
74	71, 73	syl5ibcom 248	. . . . . 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 1054	. . . . 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 620	. . . 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 18476	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → 𝐾 ∈ Poset)
78	77	ex 418	. 2 ⊢ (𝐾 ∈ 𝑉 → (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) → 𝐾 ∈ Poset))
79	22, 78	impbid2 229	1 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  {copab 5167   I cid 5545   Po wpo 5557   ↾ cres 5653  Rel wrel 5656  ‘cfv 6531  Basecbs 17367  lecple 17415  Posetcpo 18461  ltcplt 18462

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6487  df-fun 6533  df-fv 6539  df-proset 18448  df-poset 18467  df-plt 18482

This theorem is used by:  tosso  18571