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Mirrors > Home > ILE Home > Th. List > ltnqex | GIF version |
Description: The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Ref | Expression |
---|---|
ltnqex | ⊢ {𝑥 ∣ 𝑥 <Q 𝐴} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqex 7171 | . 2 ⊢ Q ∈ V | |
2 | ltrelnq 7173 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
3 | 2 | brel 4591 | . . . 4 ⊢ (𝑥 <Q 𝐴 → (𝑥 ∈ Q ∧ 𝐴 ∈ Q)) |
4 | 3 | simpld 111 | . . 3 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ Q) |
5 | 4 | abssi 3172 | . 2 ⊢ {𝑥 ∣ 𝑥 <Q 𝐴} ⊆ Q |
6 | 1, 5 | ssexi 4066 | 1 ⊢ {𝑥 ∣ 𝑥 <Q 𝐴} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 {cab 2125 Vcvv 2686 class class class wbr 3929 Qcnq 7088 <Q cltq 7093 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-qs 6435 df-ni 7112 df-nqqs 7156 df-ltnqqs 7161 |
This theorem is referenced by: nqprl 7359 nqpru 7360 1prl 7363 1pru 7364 addnqprlemrl 7365 addnqprlemru 7366 addnqprlemfl 7367 addnqprlemfu 7368 mulnqprlemrl 7381 mulnqprlemru 7382 mulnqprlemfl 7383 mulnqprlemfu 7384 ltnqpr 7401 ltnqpri 7402 archpr 7451 cauappcvgprlemladdfu 7462 cauappcvgprlemladdfl 7463 cauappcvgprlem2 7468 caucvgprlemladdfu 7485 caucvgprlem2 7488 caucvgprprlemopu 7507 suplocexprlemloc 7529 |
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