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Mirrors > Home > ILE Home > Th. List > gtnqex | GIF version |
Description: The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Ref | Expression |
---|---|
gtnqex | ⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqex 7295 | . 2 ⊢ Q ∈ V | |
2 | ltrelnq 7297 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
3 | 2 | brel 4650 | . . . 4 ⊢ (𝐴 <Q 𝑥 → (𝐴 ∈ Q ∧ 𝑥 ∈ Q)) |
4 | 3 | simprd 113 | . . 3 ⊢ (𝐴 <Q 𝑥 → 𝑥 ∈ Q) |
5 | 4 | abssi 3212 | . 2 ⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ⊆ Q |
6 | 1, 5 | ssexi 4114 | 1 ⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 {cab 2150 Vcvv 2721 class class class wbr 3976 Qcnq 7212 <Q cltq 7217 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-qs 6498 df-ni 7236 df-nqqs 7280 df-ltnqqs 7285 |
This theorem is referenced by: nqprl 7483 nqpru 7484 1prl 7487 1pru 7488 addnqprlemrl 7489 addnqprlemru 7490 addnqprlemfl 7491 addnqprlemfu 7492 mulnqprlemrl 7505 mulnqprlemru 7506 mulnqprlemfl 7507 mulnqprlemfu 7508 ltnqpr 7525 ltnqpri 7526 archpr 7575 cauappcvgprlemladdfu 7586 cauappcvgprlemladdfl 7587 cauappcvgprlem2 7592 caucvgprlemladdfu 7609 caucvgprlem2 7612 caucvgprprlemopu 7631 suplocexprlemloc 7653 |
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