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Theorem npsspw 6626
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 489 . . . 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 3393 . . . . 5 (𝑙 ∈ 𝒫 Q𝑙Q)
3 selpw 3393 . . . . 5 (𝑢 ∈ 𝒫 Q𝑢Q)
42, 3anbi12i 441 . . . 4 ((𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q) ↔ (𝑙Q𝑢Q))
51, 4sylibr 141 . . 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 4041 . 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 6621 . 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 4378 . 2 (𝒫 Q × 𝒫 Q) = {⟨𝑙, 𝑢⟩ ∣ (𝑙 ∈ 𝒫 Q𝑢 ∈ 𝒫 Q)}
96, 7, 83sstr4i 3011 1 P ⊆ (𝒫 Q × 𝒫 Q)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3a 896  wcel 1409  wral 2323  wrex 2324  wss 2944  𝒫 cpw 3386   class class class wbr 3791  {copab 3844   × cxp 4370  Qcnq 6435   <Q cltq 6440  Pcnp 6446
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-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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-opab 3846  df-xp 4378  df-inp 6621
This theorem is referenced by:  preqlu  6627  npex  6628  elinp  6629  prop  6630  elnp1st2nd  6631  cauappcvgprlemladd  6813
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