Theorem resspos 18507

Description: The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Assertion

Ref	Expression
resspos	⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)

Proof of Theorem resspos

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

Step	Hyp	Ref	Expression
1		ovexd 7454	. 2 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ V)
2		eqid 2765	. . . . . . 7 ⊢ (𝐹 ↾s 𝐴) = (𝐹 ↾s 𝐴)
3		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	2, 3	ressbas 17318	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹 ↾s 𝐴)))
5		inss2 4190	. . . . . 6 ⊢ (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
6	4, 5	eqsstrrdi 3983	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹))
7	6	adantl 487	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹))
8		eqid 2765	. . . . . . 7 ⊢ (le‘𝐹) = (le‘𝐹)
9	3, 8	ispos 18392	. . . . . 6 ⊢ (𝐹 ∈ Poset ↔ (𝐹 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
10	9	simprbi 503	. . . . 5 ⊢ (𝐹 ∈ Poset → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
11	10	adantr 486	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
12		ssralv 4007	. . . . . . . 8 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
13	12	ralimdv 3181	. . . . . . 7 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
14		ssralv 4007	. . . . . . 7 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
15	13, 14	syld 48	. . . . . 6 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
16	15	ralimdv 3181	. . . . 5 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
17		ssralv 4007	. . . . 5 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
18	16, 17	syld 48	. . . 4 ⊢ ((Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
19	7, 11, 18	sylc 66	. . 3 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
20	2, 8	ressle 17455	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (le‘𝐹) = (le‘(𝐹 ↾s 𝐴)))
21	20	adantl 487	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (le‘𝐹) = (le‘(𝐹 ↾s 𝐴)))
22		breq 5113	. . . . . . 7 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑥 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑥))
23		breq 5113	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑦 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑦))
24		breq 5113	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑥 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥))
25	23, 24	anbi12d 644	. . . . . . . 8 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) ↔ (𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥)))
26	25	imbi1d 344	. . . . . . 7 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦)))
27		breq 5113	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑧 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧))
28	23, 27	anbi12d 644	. . . . . . . 8 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) ↔ (𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧)))
29		breq 5113	. . . . . . . 8 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑧 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))
30	28, 29	imbi12d 347	. . . . . . 7 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧) ↔ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧)))
31	22, 26, 30	3anbi123d 1464	. . . . . 6 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → ((𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ (𝑥(le‘(𝐹 ↾s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))))
32	31	ralbidv 3190	. . . . 5 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘(𝐹 ↾s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))))
33	32	2ralbidv 3231	. . . 4 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘(𝐹 ↾s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))))
34	21, 33	syl 18	. . 3 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦 ∧ 𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘(𝐹 ↾s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))))
35	19, 34	mpbid 235	. 2 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘(𝐹 ↾s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧)))
36		eqid 2765	. . 3 ⊢ (Base‘(𝐹 ↾s 𝐴)) = (Base‘(𝐹 ↾s 𝐴))
37		eqid 2765	. . 3 ⊢ (le‘(𝐹 ↾s 𝐴)) = (le‘(𝐹 ↾s 𝐴))
38	36, 37	ispos 18392	. 2 ⊢ ((𝐹 ↾s 𝐴) ∈ Poset ↔ ((𝐹 ↾s 𝐴) ∈ V ∧ ∀𝑥 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝐹 ↾s 𝐴))∀𝑧 ∈ (Base‘(𝐹 ↾s 𝐴))(𝑥(le‘(𝐹 ↾s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹 ↾s 𝐴))𝑦 ∧ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧) → 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))))
39	1, 35, 38	sylanbrc 595	1 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17291   ↾_s cress 17312  lecple 17339  Posetcpo 18385

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-ple 17352  df-poset 18391

This theorem is used by:  resstos  18508