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Theorem npex 9661
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 9598 . . 3 Q ∈ V
21pwex 4766 . 2 𝒫 Q ∈ V
3 pssss 3660 . . . . 5 (𝑥Q𝑥Q)
43ad2antlr 758 . . . 4 (((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧)) → 𝑥Q)
54ss2abi 3633 . . 3 {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥𝑥Q}
6 df-np 9656 . . 3 P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
7 df-pw 4106 . . 3 𝒫 Q = {𝑥𝑥Q}
85, 6, 73sstr4i 3603 . 2 P ⊆ 𝒫 Q
92, 8ssexi 4723 1 P ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wcel 1976  {cab 2592  wral 2892  wrex 2893  Vcvv 3169  wss 3536  wpss 3537  c0 3870  𝒫 cpw 4104   class class class wbr 4574  Qcnq 9527   <Q cltq 9533  Pcnp 9534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-tr 4672  df-eprel 4936  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-om 6932  df-ni 9547  df-nq 9587  df-np 9656
This theorem is referenced by:  enrex  9741  axcnex  9821
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