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Theorem npsspw 7374
 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 519 . . . 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 velpw 3550 . . . . 5 (𝑙 ∈ 𝒫 Q𝑙Q)
3 velpw 3550 . . . . 5 (𝑢 ∈ 𝒫 Q𝑢Q)
42, 3anbi12i 456 . . . 4 ((𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q) ↔ (𝑙Q𝑢Q))
51, 4sylibr 133 . . 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 4236 . 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 7369 . 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 4589 . 2 (𝒫 Q × 𝒫 Q) = {⟨𝑙, 𝑢⟩ ∣ (𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q)}
96, 7, 83sstr4i 3169 1 P ⊆ (𝒫 Q × 𝒫 Q)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436   ⊆ wss 3102  𝒫 cpw 3543   class class class wbr 3965  {copab 4024   × cxp 4581  Qcnq 7183
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