ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elnp1st2nd GIF version

Theorem elnp1st2nd 7536
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 7531 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 3175 . . . 4 (𝐴P𝐴 ∈ (𝒫 Q × 𝒫 Q))
3 prop 7535 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 elinp 7534 . . . . . . 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 122 . . . . . 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 112 . . . . 5 (𝐴P → (((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))))
76simprd 114 . . . 4 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴)))
82, 7jca 306 . . 3 (𝐴P → (𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))))
95simprd 114 . . 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 306 . 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 6228 . . . 4 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
1211ad2antrr 488 . . 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 6218 . . . . . . . 8 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ∈ 𝒫 Q)
1413elpwid 3612 . . . . . . 7 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ⊆ Q)
15 xp2nd 6219 . . . . . . . 8 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ∈ 𝒫 Q)
1615elpwid 3612 . . . . . . 7 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ⊆ Q)
1714, 16jca 306 . . . . . 6 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → ((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q))
1817anim1i 340 . . . . 5 ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) → (((1st𝐴) ⊆ Q ∧ (2nd𝐴) ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))))
1918anim1i 340 . . . 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 134 . . 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 2270 . 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 126 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 104  wb 105  wo 709  w3a 980   = wceq 1364  wcel 2164  wral 2472  wrex 2473  wss 3153  𝒫 cpw 3601  cop 3621   class class class wbr 4029   × cxp 4657  cfv 5254  1st c1st 6191  2nd c2nd 6192  Qcnq 7340   <Q cltq 7345  Pcnp 7351
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-qs 6593  df-ni 7364  df-nqqs 7408  df-inp 7526
This theorem is referenced by:  addclpr  7597  mulclpr  7632  ltexprlempr  7668  recexprlempr  7692  cauappcvgprlemcl  7713  caucvgprlemcl  7736  caucvgprprlemcl  7764
  Copyright terms: Public domain W3C validator