Theorem resspos 32951

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 7445	. 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 17262	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹 ↾s 𝐴)))
5		inss2 4218	. . . . . 6 ⊢ (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
6	4, 5	eqsstrrdi 4009	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹))
7	6	adantl 481	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (Base‘(𝐹 ↾s 𝐴)) ⊆ (Base‘𝐹))
8		eqid 2736	. . . . . . 7 ⊢ (le‘𝐹) = (le‘𝐹)
9	3, 8	ispos 18331	. . . . . 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 496	. . . . 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 480	. . . 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 4032	. . . . . . . 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 3155	. . . . . . 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 4032	. . . . . . 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 3155	. . . . 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 4032	. . . . 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 17399	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (le‘𝐹) = (le‘(𝐹 ↾s 𝐴)))
21	20	adantl 481	. . . 4 ⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (le‘𝐹) = (le‘(𝐹 ↾s 𝐴)))
22		breq 5126	. . . . . . 7 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑥 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑥))
23		breq 5126	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑥(le‘𝐹)𝑦 ↔ 𝑥(le‘(𝐹 ↾s 𝐴))𝑦))
24		breq 5126	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑥 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑥))
25	23, 24	anbi12d 632	. . . . . . . 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 5126	. . . . . . . . 9 ⊢ ((le‘𝐹) = (le‘(𝐹 ↾s 𝐴)) → (𝑦(le‘𝐹)𝑧 ↔ 𝑦(le‘(𝐹 ↾s 𝐴))𝑧))
28	23, 27	anbi12d 632	. . . . . . . 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 5126	. . . . . . . 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 1438	. . . . . 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 3164	. . . . 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 3209	. . . 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 232	. 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 18331	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233   ↾_s cress 17256  lecple 17283  Posetcpo 18324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-dec 12714  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-ple 17296  df-poset 18330

This theorem is referenced by:  resstos  32952