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Theorem enrex 9926
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 9846 . . . 4 P ∈ V
21, 1xpex 7004 . . 3 (P × P) ∈ V
32, 2xpex 7004 . 2 ((P × P) × (P × P)) ∈ V
4 df-enr 9915 . . 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 5227 . . 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 3668 . 2 ~R ⊆ ((P × P) × (P × P))
73, 6ssexi 4836 1 ~R ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cop 4216  {copab 4745   × cxp 5141  (class class class)co 6690  Pcnp 9719   +P cpp 9721   ~R cer 9724
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  df-enr 9915
This theorem is referenced by:  addsrpr  9934  mulsrpr  9935  ltsrpr  9936  0r  9939  1sr  9940  m1r  9941  addclsr  9942  mulclsr  9943  recexsrlem  9962
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