Theorem caucvgprprlemell 7683

Description: Lemma for caucvgprpr 7710. 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 5881	. . . . . . . 8 ⊢ (𝑙 = 𝑋 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
2	1	breq2d 4015	. . . . . . 7 ⊢ (𝑙 = 𝑋 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
3	2	abbidv 2295	. . . . . 6 ⊢ (𝑙 = 𝑋 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
4	1	breq1d 4013	. . . . . . 7 ⊢ (𝑙 = 𝑋 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
5	4	abbidv 2295	. . . . . 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 3786	. . . . 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 4013	. . . 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 2478	. . 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 5518	. . . 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 7361	. . . . . 6 ⊢ Q ∈ V
12	11	rabex 4147	. . . . 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 4147	. . . . 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 6146	. . . 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 2198	. . 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 2896	. 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 3778	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑎 → ⟨𝑟, 1o⟩ = ⟨𝑎, 1o⟩)
18	17	eceq1d 6570	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑎 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
19	18	fveq2d 5519	. . . . . . . . . 10 ⊢ (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
20	19	oveq2d 5890	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
21	20	breq2d 4015	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))))
22	21	abbidv 2295	. . . . . . 7 ⊢ (𝑟 = 𝑎 → {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))})
23	20	breq1d 4013	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → ((𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞))
24	23	abbidv 2295	. . . . . . 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 3786	. . . . . 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 5515	. . . . . 6 ⊢ (𝑟 = 𝑎 → (𝐹‘𝑟) = (𝐹‘𝑎))
27	25, 26	breq12d 4016	. . . . 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 2704	. . . 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 3778	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ⟨𝑎, 1o⟩ = ⟨𝑏, 1o⟩)
30	29	eceq1d 6570	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
31	30	fveq2d 5519	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
32	31	oveq2d 5890	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
33	32	breq2d 4015	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
34	33	abbidv 2295	. . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))})
35	32	breq1d 4013	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞))
36	35	abbidv 2295	. . . . . . 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 3786	. . . . . 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 5515	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
39	37, 38	breq12d 4016	. . . . 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 2704	. . . 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 184	. . 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 457	. 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 184	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 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∃wrex 2456  {crab 2459  ⟨cop 3595   class class class wbr 4003  ‘cfv 5216  (class class class)co 5874  1^st c1st 6138  1_oc1o 6409  [cec 6532  Ncnpi 7270   ~_Q ceq 7277  Qcnq 7278   +_Q cplq 7280  *_Qcrq 7282   <_Q cltq 7283   +_P cpp 7291  <_P cltp 7293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-1st 6140  df-ec 6536  df-qs 6540  df-ni 7302  df-nqqs 7346

This theorem is referenced by:  caucvgprprlemopl  7695  caucvgprprlemlol  7696  caucvgprprlemdisj  7700  caucvgprprlemloc  7701