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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 7306 | . 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 523 | . 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 120 | 1 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∧ w3a 963 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 ⊆ wss 3076 〈cop 3535 class class class wbr 3937 Qcnq 7112 <Q cltq 7117 Pcnp 7123 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-qs 6443 df-ni 7136 df-nqqs 7180 df-inp 7298 |
This theorem is referenced by: elprnql 7313 |
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