Theorem caucvgprprlemell 7626

Description: Lemma for caucvgprpr 7653. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)

Hypothesis

Ref	Expression
caucvgprprlemell.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩

Assertion

Ref	Expression
caucvgprprlemell	⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))

Distinct variable groups:   𝐹,𝑏   𝐹,𝑙,𝑟   𝑢,𝐹,𝑟   𝑋,𝑏,𝑝   𝑋,𝑙,𝑟,𝑝   𝑢,𝑋,𝑝   𝑋,𝑞,𝑏   𝑞,𝑙,𝑟   𝑢,𝑞

Allowed substitution hints:   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑟,𝑞,𝑝,𝑏,𝑙)

Proof of Theorem caucvgprprlemell

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5849	. . . . . . . 8 ⊢ (𝑙 = 𝑋 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
2	1	breq2d 3994	. . . . . . 7 ⊢ (𝑙 = 𝑋 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
3	2	abbidv 2284	. . . . . 6 ⊢ (𝑙 = 𝑋 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
4	1	breq1d 3992	. . . . . . 7 ⊢ (𝑙 = 𝑋 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
5	4	abbidv 2284	. . . . . 6 ⊢ (𝑙 = 𝑋 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞})
6	3, 5	opeq12d 3766	. . . . 5 ⊢ (𝑙 = 𝑋 → ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩)
7	6	breq1d 3992	. . . 4 ⊢ (𝑙 = 𝑋 → (⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
8	7	rexbidv 2467	. . 3 ⊢ (𝑙 = 𝑋 → (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
9		caucvgprprlemell.lim	. . . . 5 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
10	9	fveq2i 5489	. . . 4 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩)
11		nqex 7304	. . . . . 6 ⊢ Q ∈ V
12	11	rabex 4126	. . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)} ∈ V
13	11	rabex 4126	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩} ∈ V
14	12, 13	op1st 6114	. . . 4 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩) = {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}
15	10, 14	eqtri 2186	. . 3 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}
16	8, 15	elrab2 2885	. 2 ⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
17		opeq1 3758	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑎 → ⟨𝑟, 1o⟩ = ⟨𝑎, 1o⟩)
18	17	eceq1d 6537	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑎 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
19	18	fveq2d 5490	. . . . . . . . . 10 ⊢ (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
20	19	oveq2d 5858	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
21	20	breq2d 3994	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))))
22	21	abbidv 2284	. . . . . . 7 ⊢ (𝑟 = 𝑎 → {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))})
23	20	breq1d 3992	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → ((𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞))
24	23	abbidv 2284	. . . . . . 7 ⊢ (𝑟 = 𝑎 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞})
25	22, 24	opeq12d 3766	. . . . . 6 ⊢ (𝑟 = 𝑎 → ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩)
26		fveq2 5486	. . . . . 6 ⊢ (𝑟 = 𝑎 → (𝐹‘𝑟) = (𝐹‘𝑎))
27	25, 26	breq12d 3995	. . . . 5 ⊢ (𝑟 = 𝑎 → (⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎)))
28	27	cbvrexv 2693	. . . 4 ⊢ (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑎 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎))
29		opeq1 3758	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ⟨𝑎, 1o⟩ = ⟨𝑏, 1o⟩)
30	29	eceq1d 6537	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
31	30	fveq2d 5490	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
32	31	oveq2d 5858	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
33	32	breq2d 3994	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
34	33	abbidv 2284	. . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))})
35	32	breq1d 3992	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞))
36	35	abbidv 2284	. . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞})
37	34, 36	opeq12d 3766	. . . . . 6 ⊢ (𝑎 = 𝑏 → ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩)
38		fveq2 5486	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
39	37, 38	breq12d 3995	. . . . 5 ⊢ (𝑎 = 𝑏 → (⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
40	39	cbvrexv 2693	. . . 4 ⊢ (∃𝑎 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎) ↔ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
41	28, 40	bitri 183	. . 3 ⊢ (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
42	41	anbi2i 453	. 2 ⊢ ((𝑋 ∈ Q ∧ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
43	16, 42	bitri 183	1 ⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {cab 2151  ∃wrex 2445  {crab 2448  ⟨cop 3579   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  1_oc1o 6377  [cec 6499  Ncnpi 7213   ~_Q ceq 7220  Qcnq 7221   +_Q cplq 7223  *_Qcrq 7225   <_Q cltq 7226   +_P cpp 7234  <_P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-1st 6108  df-ec 6503  df-qs 6507  df-ni 7245  df-nqqs 7289

This theorem is referenced by:  caucvgprprlemopl  7638  caucvgprprlemlol  7639  caucvgprprlemdisj  7643  caucvgprprlemloc  7644