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Theorem npex 10929
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 10866 . . 3 Q ∈ V
21pwex 5340 . 2 𝒫 Q ∈ V
3 pssss 4060 . . . . 5 (𝑥Q𝑥Q)
43ad2antlr 726 . . . 4 (((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧)) → 𝑥Q)
54ss2abi 4028 . . 3 {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))} ⊆ {𝑥𝑥Q}
6 df-np 10924 . . 3 P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
7 df-pw 4567 . . 3 𝒫 Q = {𝑥𝑥Q}
85, 6, 73sstr4i 3992 . 2 P ⊆ 𝒫 Q
92, 8ssexi 5284 1 P ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  {cab 2714  wral 3065  wrex 3074  Vcvv 3448  wss 3915  wpss 3916  c0 4287  𝒫 cpw 4565   class class class wbr 5110  Qcnq 10795   <Q cltq 10801  Pcnp 10802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-om 7808  df-ni 10815  df-nq 10855  df-np 10924
This theorem is referenced by:  nrex1  11007  enrex  11010
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