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Theorem nqprrnd 7373
 Description: A cut produced from a rational is rounded. Lemma for nqprlu 7377. (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
nqprrnd (𝐴Q → (∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))))
Distinct variable group:   𝑥,𝐴,𝑟,𝑞

Proof of Theorem nqprrnd
StepHypRef Expression
1 ltbtwnnqq 7245 . . . . . 6 (𝐴 <Q 𝑟 ↔ ∃𝑞Q (𝐴 <Q 𝑞𝑞 <Q 𝑟))
2 ancom 264 . . . . . . 7 ((𝐴 <Q 𝑞𝑞 <Q 𝑟) ↔ (𝑞 <Q 𝑟𝐴 <Q 𝑞))
32rexbii 2445 . . . . . 6 (∃𝑞Q (𝐴 <Q 𝑞𝑞 <Q 𝑟) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝐴 <Q 𝑞))
41, 3bitri 183 . . . . 5 (𝐴 <Q 𝑟 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝐴 <Q 𝑞))
5 vex 2692 . . . . . 6 𝑟 ∈ V
6 breq2 3939 . . . . . 6 (𝑥 = 𝑟 → (𝐴 <Q 𝑥𝐴 <Q 𝑟))
75, 6elab 2831 . . . . 5 (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟)
8 vex 2692 . . . . . . . 8 𝑞 ∈ V
9 breq2 3939 . . . . . . . 8 (𝑥 = 𝑞 → (𝐴 <Q 𝑥𝐴 <Q 𝑞))
108, 9elab 2831 . . . . . . 7 (𝑞 ∈ {𝑥𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑞)
1110anbi2i 453 . . . . . 6 ((𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}) ↔ (𝑞 <Q 𝑟𝐴 <Q 𝑞))
1211rexbii 2445 . . . . 5 (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝐴 <Q 𝑞))
134, 7, 123bitr4i 211 . . . 4 (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))
1413rgenw 2490 . . 3 𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))
1514a1i 9 . 2 (𝐴Q → ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥})))
16 ltbtwnnqq 7245 . . . 4 (𝑞 <Q 𝐴 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 <Q 𝐴))
17 breq1 3938 . . . . 5 (𝑥 = 𝑞 → (𝑥 <Q 𝐴𝑞 <Q 𝐴))
188, 17elab 2831 . . . 4 (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴)
19 breq1 3938 . . . . . . 7 (𝑥 = 𝑟 → (𝑥 <Q 𝐴𝑟 <Q 𝐴))
205, 19elab 2831 . . . . . 6 (𝑟 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝑟 <Q 𝐴)
2120anbi2i 453 . . . . 5 ((𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴}) ↔ (𝑞 <Q 𝑟𝑟 <Q 𝐴))
2221rexbii 2445 . . . 4 (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴}) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 <Q 𝐴))
2316, 18, 223bitr4i 211 . . 3 (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴}))
2423rgenw 2490 . 2 𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴}))
2515, 24jctil 310 1 (𝐴Q → (∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418   class class class wbr 3935  Qcnq 7110
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