Theorem resspos 31826

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 7392	. 2 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ V)
2		eqid 2736	. . . . . . 7 ⊢ (𝐹 ↾s 𝐴) = (𝐹 ↾s 𝐴)
3		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	2, 3	ressbas 17118	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹 ↾s 𝐴)))
5		inss2 4189	. . . . . 6 ⊢ (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
6	4, 5	eqsstrrdi 3999	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹))
7	6	adantl 482	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹))
8		eqid 2736	. . . . . . 7 ⊢ (le‘𝐹) = (le‘𝐹)
9	3, 8	ispos 18203	. . . . . 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 497	. . . . 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 481	. . . 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 4010	. . . . . . . 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 3166	. . . . . . 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 4010	. . . . . . 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 47	. . . . . 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 3166	. . . . 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 4010	. . . . 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 47	. . . 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 65	. . 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 17261	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (le‘𝐹) = (le‘(𝐹 ↾s 𝐴)))
21	20	adantl 482	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (le‘𝐹) = (le‘(𝐹 ↾s 𝐴)))
22		breq 5107	. . . . . . 7 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑥 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑥))
23		breq 5107	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑦 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑦))
24		breq 5107	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑥 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥))
25	23, 24	anbi12d 631	. . . . . . . 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 341	. . . . . . 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 5107	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑧 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧))
28	23, 27	anbi12d 631	. . . . . . . 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 5107	. . . . . . . 8 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑧 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))
30	28, 29	imbi12d 344	. . . . . . 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 1436	. . . . . 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 3174	. . . . 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 3212	. . . 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 17	. . 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 231	. 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 2736	. . 3 ⊢ (Base‘(𝐹 ↾s 𝐴)) = (Base‘(𝐹 ↾s 𝐴))
37		eqid 2736	. . 3 ⊢ (le‘(𝐹 ↾s 𝐴)) = (le‘(𝐹 ↾s 𝐴))
38	36, 37	ispos 18203	. 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 583	1 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083   ↾_s cress 17112  lecple 17140  Posetcpo 18196

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-ple 17153  df-poset 18202

This theorem is referenced by:  resstos  31827