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Theorem npex 10977
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 10914 . . 3 Q ∈ V
21pwex 5377 . 2 𝒫 Q ∈ V
3 pssss 4094 . . . . 5 (𝑥Q𝑥Q)
43ad2antlr 725 . . . 4 (((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧)) → 𝑥Q)
54ss2abi 4062 . . 3 {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥𝑥Q}
6 df-np 10972 . . 3 P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
7 df-pw 4603 . . 3 𝒫 Q = {𝑥𝑥Q}
85, 6, 73sstr4i 4024 . 2 P ⊆ 𝒫 Q
92, 8ssexi 5321 1 P ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539  wcel 2106  {cab 2709  wral 3061  wrex 3070  Vcvv 3474  wss 3947  wpss 3948  c0 4321  𝒫 cpw 4601   class class class wbr 5147  Qcnq 10843   <Q cltq 10849  Pcnp 10850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7852  df-ni 10863  df-nq 10903  df-np 10972
This theorem is referenced by:  nrex1  11055  enrex  11058
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