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Theorem elnp1st2nd 6632
Description: Membership in positive reals, using 1st and 2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
Assertion
Ref Expression
elnp1st2nd (𝐴P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))
Distinct variable group:   𝑟,𝑞,𝐴

Proof of Theorem elnp1st2nd
StepHypRef Expression
1 npsspw 6627 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 2969 . . . 4 (𝐴P𝐴 ∈ (𝒫 Q × 𝒫 Q))
3 prop 6631 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 elinp 6630 . . . . . . 7 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ↔ ((((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))
53, 4sylib 131 . . . . . 6 (𝐴P → ((((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))
65simpld 109 . . . . 5 (𝐴P → (((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))))
76simprd 111 . . . 4 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴)))
82, 7jca 294 . . 3 (𝐴P → (𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))))
95simprd 111 . . 3 (𝐴P → ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴)))))
108, 9jca 294 . 2 (𝐴P → ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))
11 1st2nd2 5829 . . . 4 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
1211ad2antrr 465 . . 3 (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
13 xp1st 5820 . . . . . . . 8 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ∈ 𝒫 Q)
1413elpwid 3397 . . . . . . 7 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ⊆ Q)
15 xp2nd 5821 . . . . . . . 8 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ∈ 𝒫 Q)
1615elpwid 3397 . . . . . . 7 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ⊆ Q)
1714, 16jca 294 . . . . . 6 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → ((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q))
1817anim1i 327 . . . . 5 ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) → (((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))))
1918anim1i 327 . . . 4 (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))) → ((((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))
2019, 4sylibr 141 . . 3 (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2112, 20eqeltrd 2130 . 2 (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))) → 𝐴P)
2210, 21impbii 121 1 (𝐴P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3a 896   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2945  𝒫 cpw 3387  cop 3406   class class class wbr 3792   × cxp 4371  cfv 4930  1st c1st 5793  2nd c2nd 5794  Qcnq 6436   <Q cltq 6441  Pcnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-1st 5795  df-2nd 5796  df-qs 6143  df-ni 6460  df-nqqs 6504  df-inp 6622
This theorem is referenced by:  addclpr  6693  mulclpr  6728  ltexprlempr  6764  recexprlempr  6788  cauappcvgprlemcl  6809  caucvgprlemcl  6832  caucvgprprlemcl  6860
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