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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 7340 | . 2 ⊢ Q ∈ V | |
2 | ltrelnq 7342 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
3 | 2 | brel 4674 | . . . 4 ⊢ (𝐴 <Q 𝑥 → (𝐴 ∈ Q ∧ 𝑥 ∈ Q)) |
4 | 3 | simprd 114 | . . 3 ⊢ (𝐴 <Q 𝑥 → 𝑥 ∈ Q) |
5 | 4 | abssi 3230 | . 2 ⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ⊆ Q |
6 | 1, 5 | ssexi 4138 | 1 ⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 {cab 2163 Vcvv 2737 class class class wbr 4000 Qcnq 7257 <Q cltq 7262 |
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 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-iinf 4583 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-qs 6534 df-ni 7281 df-nqqs 7325 df-ltnqqs 7330 |
This theorem is referenced by: nqprl 7528 nqpru 7529 1prl 7532 1pru 7533 addnqprlemrl 7534 addnqprlemru 7535 addnqprlemfl 7536 addnqprlemfu 7537 mulnqprlemrl 7550 mulnqprlemru 7551 mulnqprlemfl 7552 mulnqprlemfu 7553 ltnqpr 7570 ltnqpri 7571 archpr 7620 cauappcvgprlemladdfu 7631 cauappcvgprlemladdfl 7632 cauappcvgprlem2 7637 caucvgprlemladdfu 7654 caucvgprlem2 7657 caucvgprprlemopu 7676 suplocexprlemloc 7698 |
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