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Mirrors > Home > ILE Home > Th. List > prssnql | GIF version |
Description: The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
prssnql | ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinp 7504 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P ↔ (((𝐿 ⊆ Q ∧ 𝑈 ⊆ Q) ∧ (∃𝑥 ∈ Q 𝑥 ∈ 𝐿 ∧ ∃𝑦 ∈ Q 𝑦 ∈ 𝑈)) ∧ ((∀𝑥 ∈ Q (𝑥 ∈ 𝐿 ↔ ∃𝑦 ∈ Q (𝑥 <Q 𝑦 ∧ 𝑦 ∈ 𝐿)) ∧ ∀𝑦 ∈ Q (𝑦 ∈ 𝑈 ↔ ∃𝑥 ∈ Q (𝑥 <Q 𝑦 ∧ 𝑥 ∈ 𝑈))) ∧ ∀𝑥 ∈ Q ¬ (𝑥 ∈ 𝐿 ∧ 𝑥 ∈ 𝑈) ∧ ∀𝑥 ∈ Q ∀𝑦 ∈ Q (𝑥 <Q 𝑦 → (𝑥 ∈ 𝐿 ∨ 𝑦 ∈ 𝑈))))) | |
2 | simplll 533 | . 2 ⊢ ((((𝐿 ⊆ Q ∧ 𝑈 ⊆ Q) ∧ (∃𝑥 ∈ Q 𝑥 ∈ 𝐿 ∧ ∃𝑦 ∈ Q 𝑦 ∈ 𝑈)) ∧ ((∀𝑥 ∈ Q (𝑥 ∈ 𝐿 ↔ ∃𝑦 ∈ Q (𝑥 <Q 𝑦 ∧ 𝑦 ∈ 𝐿)) ∧ ∀𝑦 ∈ Q (𝑦 ∈ 𝑈 ↔ ∃𝑥 ∈ Q (𝑥 <Q 𝑦 ∧ 𝑥 ∈ 𝑈))) ∧ ∀𝑥 ∈ Q ¬ (𝑥 ∈ 𝐿 ∧ 𝑥 ∈ 𝑈) ∧ ∀𝑥 ∈ Q ∀𝑦 ∈ Q (𝑥 <Q 𝑦 → (𝑥 ∈ 𝐿 ∨ 𝑦 ∈ 𝑈)))) → 𝐿 ⊆ Q) | |
3 | 1, 2 | sylbi 121 | 1 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∧ w3a 980 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 ⊆ wss 3144 〈cop 3610 class class class wbr 4018 Qcnq 7310 <Q cltq 7315 Pcnp 7321 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-qs 6566 df-ni 7334 df-nqqs 7378 df-inp 7496 |
This theorem is referenced by: elprnql 7511 |
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