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Theorem npsspw 6569
 Description: Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
Assertion
Ref Expression
npsspw P ⊆ (𝒫 Q × 𝒫 Q)

Proof of Theorem npsspw
Dummy variables 𝑢 𝑙 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 481 . . . 4 ((((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢)))) → (𝑙Q𝑢Q))
2 selpw 3366 . . . . 5 (𝑙 ∈ 𝒫 Q𝑙Q)
3 selpw 3366 . . . . 5 (𝑢 ∈ 𝒫 Q𝑢Q)
42, 3anbi12i 433 . . . 4 ((𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q) ↔ (𝑙Q𝑢Q))
51, 4sylibr 137 . . 3 ((((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢)))) → (𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q))
65ssopab2i 4014 . 2 {⟨𝑙, 𝑢⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))} ⊆ {⟨𝑙, 𝑢⟩ ∣ (𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q)}
7 df-inp 6564 . 2 P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))}
8 df-xp 4351 . 2 (𝒫 Q × 𝒫 Q) = {⟨𝑙, 𝑢⟩ ∣ (𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q)}
96, 7, 83sstr4i 2984 1 P ⊆ (𝒫 Q × 𝒫 Q)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  𝒫 cpw 3359   class class class wbr 3764  {copab 3817   × cxp 4343  Qcnq 6378
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