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Theorem npex 9846
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 9783 . . 3 Q ∈ V
21pwex 4878 . 2 𝒫 Q ∈ V
3 pssss 3735 . . . . 5 (𝑥Q𝑥Q)
43ad2antlr 763 . . . 4 (((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧)) → 𝑥Q)
54ss2abi 3707 . . 3 {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥𝑥Q}
6 df-np 9841 . . 3 P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
7 df-pw 4193 . . 3 𝒫 Q = {𝑥𝑥Q}
85, 6, 73sstr4i 3677 . 2 P ⊆ 𝒫 Q
92, 8ssexi 4836 1 P ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  wss 3607  wpss 3608  c0 3948  𝒫 cpw 4191   class class class wbr 4685  Qcnq 9712   <Q cltq 9718  Pcnp 9719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108  df-ni 9732  df-nq 9772  df-np 9841
This theorem is referenced by:  enrex  9926  axcnex  10006
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