Theorem pospo 18280

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 18270	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 < 𝑥)
3		pospo.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4	3, 1	plttr 18277	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧))
5	2, 4	ispod 5590	. . 3 ⊢ (𝐾 ∈ Poset → < Po 𝐵)
6		relres 6002	. . . . 5 ⊢ Rel ( I ↾ 𝐵)
7	6	a1i 11	. . . 4 ⊢ (𝐾 ∈ Poset → Rel ( I ↾ 𝐵))
8		opabresid 6039	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)}
9	8	eqcomi 2740	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐵)
10	9	eleq2i 2824	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝑥)} ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵))
11		opabidw 5517	. . . . . 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 18253	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
15		df-br 5142	. . . . . . . 8 ⊢ (𝑥 ≤ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≤ )
16		breq2 5145	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑥))
17	15, 16	bitr3id 284	. . . . . . 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	biimtrid 241	. . . 4 ⊢ (𝐾 ∈ Poset → (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ≤ ))
21	7, 20	relssdv 5780	. . 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 5984	. . . . . . 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 776	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵) → ( I ↾ 𝐵) ⊆ ≤ )
32	31	ssbrd 5184	. . . . 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 18271	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
35	34	3adant1 1130	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
36	3, 13, 1	pleval2i 18271	. . . . . . 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 1130	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
39		simprl 769	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) → < Po 𝐵)
40		po2nr 5595	. . . . . . . . 9 ⊢ (( < Po 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
41	40	3impb 1115	. . . . . . . 8 ⊢ (( < Po 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ¬ (𝑥 < 𝑦 ∧ 𝑦 < 𝑥))
42	39, 41	syl3an1 1163	. . . . . . 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 1037	. . . . 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 1194	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
54		simpr2 1195	. . . . . 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 1196	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
57	3, 13, 1	pleval2i 18271	. . . . . 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 5594	. . . . . . . 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 765	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐾 ∈ 𝑉)
62	13, 1	pltle 18268	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 < 𝑧 → 𝑥 ≤ 𝑧))
63	61, 53, 56, 62	syl3anc 1371	. . . . . . 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 5144	. . . . . . . 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 5145	. . . . . . . 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 2754	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑧)
73	72	breq2d 5153	. . . . . . 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 1037	. . . . 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 18258	. . 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 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ⊆ wss 3944  ⟨cop 4628   class class class wbr 5141  {copab 5203   I cid 5566   Po wpo 5579   ↾ cres 5671  Rel wrel 5674  ‘cfv 6532  Basecbs 17126  lecple 17186  Posetcpo 18242  ltcplt 18243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6484  df-fun 6534  df-fv 6540  df-proset 18230  df-poset 18248  df-plt 18265

This theorem is referenced by:  tosso  18354