Theorem brecop 6132

Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Hypotheses

Ref	Expression
brecop.1	⊢ ∼ ∈ V
brecop.2	⊢ ∼ Er (𝐺 × 𝐺)
brecop.4	⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
brecop.5	⊢ ≤ = {⟨x, y⟩ ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ))}
brecop.6	⊢ ((((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (A ∈ 𝐺 ∧ B ∈ 𝐺)) ∧ ((v ∈ 𝐺 ∧ u ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (φ ↔ ψ)))

Assertion

Ref	Expression
brecop	⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨A, B⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ ψ))

Distinct variable groups:   x,y,z,w,v,u,A   x,B,y,z,w,v,u   x,𝐶,y,z,w,v,u   x,𝐷,y,z,w,v,u   x, ∼ ,y,z,w,v,u   x,𝐻,y   z,𝐺,w,v,u   φ,x,y   ψ,z,w,v,u

Allowed substitution hints:   φ(z,w,v,u)   ψ(x,y)   𝐺(x,y)   𝐻(z,w,v,u)   ≤ (x,y,z,w,v,u)

Proof of Theorem brecop

Step	Hyp	Ref	Expression
1		brecop.1	. . . 4 ⊢ ∼ ∈ V
2		brecop.4	. . . 4 ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
3	1, 2	ecopqsi 6097	. . 3 ⊢ ((A ∈ 𝐺 ∧ B ∈ 𝐺) → [⟨A, B⟩] ∼ ∈ 𝐻)
4	1, 2	ecopqsi 6097	. . 3 ⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻)
5		df-br 3756	. . . . 5 ⊢ ([⟨A, B⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ ⟨[⟨A, B⟩] ∼ , [⟨𝐶, 𝐷⟩] ∼ ⟩ ∈ ≤ )
6		brecop.5	. . . . . 6 ⊢ ≤ = {⟨x, y⟩ ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ))}
7	6	eleq2i 2101	. . . . 5 ⊢ (⟨[⟨A, B⟩] ∼ , [⟨𝐶, 𝐷⟩] ∼ ⟩ ∈ ≤ ↔ ⟨[⟨A, B⟩] ∼ , [⟨𝐶, 𝐷⟩] ∼ ⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ))})
8	5, 7	bitri 173	. . . 4 ⊢ ([⟨A, B⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ ⟨[⟨A, B⟩] ∼ , [⟨𝐶, 𝐷⟩] ∼ ⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ))})
9		eqeq1 2043	. . . . . . . 8 ⊢ (x = [⟨A, B⟩] ∼ → (x = [⟨z, w⟩] ∼ ↔ [⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ))
10	9	anbi1d 438	. . . . . . 7 ⊢ (x = [⟨A, B⟩] ∼ → ((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ↔ ([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ )))
11	10	anbi1d 438	. . . . . 6 ⊢ (x = [⟨A, B⟩] ∼ → (((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ) ↔ (([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ)))
12	11	4exbidv 1747	. . . . 5 ⊢ (x = [⟨A, B⟩] ∼ → (∃z∃w∃v∃u((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ) ↔ ∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ)))
13		eqeq1 2043	. . . . . . . 8 ⊢ (y = [⟨𝐶, 𝐷⟩] ∼ → (y = [⟨v, u⟩] ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ))
14	13	anbi2d 437	. . . . . . 7 ⊢ (y = [⟨𝐶, 𝐷⟩] ∼ → (([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ↔ ([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ )))
15	14	anbi1d 438	. . . . . 6 ⊢ (y = [⟨𝐶, 𝐷⟩] ∼ → ((([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ) ↔ (([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ)))
16	15	4exbidv 1747	. . . . 5 ⊢ (y = [⟨𝐶, 𝐷⟩] ∼ → (∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ) ↔ ∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ)))
17	12, 16	opelopab2 3998	. . . 4 ⊢ (([⟨A, B⟩] ∼ ∈ 𝐻 ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻) → (⟨[⟨A, B⟩] ∼ , [⟨𝐶, 𝐷⟩] ∼ ⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ∼ ∧ y = [⟨v, u⟩] ∼ ) ∧ φ))} ↔ ∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ)))
18	8, 17	syl5bb 181	. . 3 ⊢ (([⟨A, B⟩] ∼ ∈ 𝐻 ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻) → ([⟨A, B⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ ∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ)))
19	3, 4, 18	syl2an 273	. 2 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨A, B⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ ∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ)))
20		opeq12 3542	. . . . . 6 ⊢ ((z = A ∧ w = B) → ⟨z, w⟩ = ⟨A, B⟩)
21	20	eceq1d 6078	. . . . 5 ⊢ ((z = A ∧ w = B) → [⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ )
22		opeq12 3542	. . . . . 6 ⊢ ((v = 𝐶 ∧ u = 𝐷) → ⟨v, u⟩ = ⟨𝐶, 𝐷⟩)
23	22	eceq1d 6078	. . . . 5 ⊢ ((v = 𝐶 ∧ u = 𝐷) → [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
24	21, 23	anim12i 321	. . . 4 ⊢ (((z = A ∧ w = B) ∧ (v = 𝐶 ∧ u = 𝐷)) → ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ))
25		opelxpi 4319	. . . . . . . 8 ⊢ ((A ∈ 𝐺 ∧ B ∈ 𝐺) → ⟨A, B⟩ ∈ (𝐺 × 𝐺))
26		opelxp 4317	. . . . . . . . 9 ⊢ (⟨z, w⟩ ∈ (𝐺 × 𝐺) ↔ (z ∈ 𝐺 ∧ w ∈ 𝐺))
27		brecop.2	. . . . . . . . . . 11 ⊢ ∼ Er (𝐺 × 𝐺)
28	27	a1i 9	. . . . . . . . . 10 ⊢ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ → ∼ Er (𝐺 × 𝐺))
29		id 19	. . . . . . . . . 10 ⊢ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ → [⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ )
30	28, 29	ereldm 6085	. . . . . . . . 9 ⊢ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ → (⟨z, w⟩ ∈ (𝐺 × 𝐺) ↔ ⟨A, B⟩ ∈ (𝐺 × 𝐺)))
31	26, 30	syl5bbr 183	. . . . . . . 8 ⊢ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ → ((z ∈ 𝐺 ∧ w ∈ 𝐺) ↔ ⟨A, B⟩ ∈ (𝐺 × 𝐺)))
32	25, 31	syl5ibr 145	. . . . . . 7 ⊢ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ → ((A ∈ 𝐺 ∧ B ∈ 𝐺) → (z ∈ 𝐺 ∧ w ∈ 𝐺)))
33		opelxpi 4319	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
34		opelxp 4317	. . . . . . . . 9 ⊢ (⟨v, u⟩ ∈ (𝐺 × 𝐺) ↔ (v ∈ 𝐺 ∧ u ∈ 𝐺))
35	27	a1i 9	. . . . . . . . . 10 ⊢ ([⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ → ∼ Er (𝐺 × 𝐺))
36		id 19	. . . . . . . . . 10 ⊢ ([⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ → [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
37	35, 36	ereldm 6085	. . . . . . . . 9 ⊢ ([⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ → (⟨v, u⟩ ∈ (𝐺 × 𝐺) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺)))
38	34, 37	syl5bbr 183	. . . . . . . 8 ⊢ ([⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ → ((v ∈ 𝐺 ∧ u ∈ 𝐺) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺)))
39	33, 38	syl5ibr 145	. . . . . . 7 ⊢ ([⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ → ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → (v ∈ 𝐺 ∧ u ∈ 𝐺)))
40	32, 39	im2anan9 530	. . . . . 6 ⊢ (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (v ∈ 𝐺 ∧ u ∈ 𝐺))))
41		brecop.6	. . . . . . . . 9 ⊢ ((((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (A ∈ 𝐺 ∧ B ∈ 𝐺)) ∧ ((v ∈ 𝐺 ∧ u ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (φ ↔ ψ)))
42	41	an4s 522	. . . . . . . 8 ⊢ ((((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (v ∈ 𝐺 ∧ u ∈ 𝐺)) ∧ ((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (φ ↔ ψ)))
43	42	ex 108	. . . . . . 7 ⊢ (((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (v ∈ 𝐺 ∧ u ∈ 𝐺)) → (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (φ ↔ ψ))))
44	43	com13 74	. . . . . 6 ⊢ (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (v ∈ 𝐺 ∧ u ∈ 𝐺)) → (φ ↔ ψ))))
45	40, 44	mpdd 36	. . . . 5 ⊢ (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (φ ↔ ψ)))
46	45	pm5.74d 171	. . . 4 ⊢ (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → ((((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ) ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ψ)))
47	24, 46	cgsex4g 2585	. . 3 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃z∃w∃v∃u(([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ)) ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ψ)))
48		eqcom 2039	. . . . . . 7 ⊢ ([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ↔ [⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ )
49		eqcom 2039	. . . . . . 7 ⊢ ([⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ↔ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
50	48, 49	anbi12i 433	. . . . . 6 ⊢ (([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ↔ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ))
51	50	a1i 9	. . . . 5 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ↔ ([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )))
52		biimt 230	. . . . 5 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (φ ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ)))
53	51, 52	anbi12d 442	. . . 4 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ((([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ) ↔ (([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ))))
54	53	4exbidv 1747	. . 3 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ) ↔ ∃z∃w∃v∃u(([⟨z, w⟩] ∼ = [⟨A, B⟩] ∼ ∧ [⟨v, u⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ))))
55		biimt 230	. . 3 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (ψ ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ψ)))
56	47, 54, 55	3bitr4d 209	. 2 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃z∃w∃v∃u(([⟨A, B⟩] ∼ = [⟨z, w⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨v, u⟩] ∼ ) ∧ φ) ↔ ψ))
57	19, 56	bitrd 177	1 ⊢ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨A, B⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286   Er wer 6039  [cec 6040   / cqs 6041

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044  df-qs 6048

This theorem is referenced by:  ordpipqqs  6358  ltsrprg  6675