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Theorem npex 11024
Description: The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
Assertion
Ref Expression
npex P ∈ V

Proof of Theorem npex
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqex 10961 . . 3 Q ∈ V
21pwex 5386 . 2 𝒫 Q ∈ V
3 pssss 4108 . . . . 5 (𝑥Q𝑥Q)
43ad2antlr 727 . . . 4 (((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧)) → 𝑥Q)
54ss2abi 4077 . . 3 {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥𝑥Q}
6 df-np 11019 . . 3 P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
7 df-pw 4607 . . 3 𝒫 Q = {𝑥𝑥Q}
85, 6, 73sstr4i 4039 . 2 P ⊆ 𝒫 Q
92, 8ssexi 5328 1 P ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  wss 3963  wpss 3964  c0 4339  𝒫 cpw 4605   class class class wbr 5148  Qcnq 10890   <Q cltq 10896  Pcnp 10897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888  df-ni 10910  df-nq 10950  df-np 11019
This theorem is referenced by:  nrex1  11102  enrex  11105
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