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Theorem enrex 11058
Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
Assertion
Ref Expression
enrex ~R ∈ V

Proof of Theorem enrex
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 npex 10977 . . . 4 P ∈ V
21, 1xpex 7736 . . 3 (P × P) ∈ V
32, 2xpex 7736 . 2 ((P × P) × (P × P)) ∈ V
4 df-enr 11046 . . 3 ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
5 opabssxp 5766 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} ⊆ ((P × P) × (P × P))
64, 5eqsstri 4015 . 2 ~R ⊆ ((P × P) × (P × P))
73, 6ssexi 5321 1 ~R ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  cop 4633  {copab 5209   × cxp 5673  (class class class)co 7405  Pcnp 10850   +P cpp 10852   ~R cer 10855
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  df-enr 11046
This theorem is referenced by:  addsrpr  11066  mulsrpr  11067  ltsrpr  11068  0r  11071  1sr  11072  m1r  11073  addclsr  11074  mulclsr  11075  recexsrlem  11094
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