Theorem resspos 18386

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 7391	. 2 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ V)
2		eqid 2739	. . . . . . 7 ⊢ (𝐹 ↾s 𝐴) = (𝐹 ↾s 𝐴)
3		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹)
4	2, 3	ressbas 17197	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹 ↾s 𝐴)))
5		inss2 4166	. . . . . 6 ⊢ (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
6	4, 5	eqsstrrdi 3960	. . . . 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 2739	. . . . . . 7 ⊢ (le‘𝐹) = (le‘𝐹)
9	3, 8	ispos 18271	. . . . . 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 498	. . . . 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 3983	. . . . . . . 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 3153	. . . . . . 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 3983	. . . . . . 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 3153	. . . . 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 3983	. . . . 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 17334	. . . . 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 5074	. . . . . . 7 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑥 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑥))
23		breq 5074	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑦 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑦))
24		breq 5074	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑥 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥))
25	23, 24	anbi12d 638	. . . . . . . 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 342	. . . . . . 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 5074	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑧 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧))
28	23, 27	anbi12d 638	. . . . . . . 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 5074	. . . . . . . 8 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑧 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑧))
30	28, 29	imbi12d 345	. . . . . . 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 1444	. . . . . 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 3162	. . . . 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 3203	. . . 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 233	. 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 2739	. . 3 ⊢ (Base‘(𝐹 ↾s 𝐴)) = (Base‘(𝐹 ↾s 𝐴))
37		eqid 2739	. . 3 ⊢ (le‘(𝐹 ↾s 𝐴)) = (le‘(𝐹 ↾s 𝐴))
38	36, 37	ispos 18271	. 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 589	1 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170   ↾_s cress 17191  lecple 17218  Posetcpo 18264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-dec 12636  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-ple 17231  df-poset 18270

This theorem is referenced by:  resstos  18387