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Theorem npsspw 6319
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 469 . . . 4 ((((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u)))) → (𝑙Q uQ))
2 selpw 3337 . . . . 5 (𝑙 𝒫 Q𝑙Q)
3 selpw 3337 . . . . 5 (u 𝒫 QuQ)
42, 3anbi12i 436 . . . 4 ((𝑙 𝒫 Q u 𝒫 Q) ↔ (𝑙Q uQ))
51, 4sylibr 137 . . 3 ((((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u)))) → (𝑙 𝒫 Q u 𝒫 Q))
65ssopab2i 3984 . 2 {⟨𝑙, u⟩ ∣ (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))} ⊆ {⟨𝑙, u⟩ ∣ (𝑙 𝒫 Q u 𝒫 Q)}
7 df-inp 6314 . 2 P = {⟨𝑙, u⟩ ∣ (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))}
8 df-xp 4274 . 2 (𝒫 Q × 𝒫 Q) = {⟨𝑙, u⟩ ∣ (𝑙 𝒫 Q u 𝒫 Q)}
96, 7, 83sstr4i 2957 1 P ⊆ (𝒫 Q × 𝒫 Q)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616   w3a 871   wcel 1370  wral 2280  wrex 2281  wss 2890  𝒫 cpw 3330   class class class wbr 3734  {copab 3787   × cxp 4266  Qcnq 6134   <Q cltq 6139  Pcnp 6145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332  df-opab 3789  df-xp 4274  df-inp 6314
This theorem is referenced by:  preqlu  6320  npex  6321  elinp  6322  prop  6323  elnp1st2nd  6324
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