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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
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