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Theorem enrex 10089
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 10009 . . . 4 P ∈ V
21, 1xpex 7108 . . 3 (P × P) ∈ V
32, 2xpex 7108 . 2 ((P × P) × (P × P)) ∈ V
4 df-enr 10078 . . 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 5333 . . 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 3782 . 2 ~R ⊆ ((P × P) × (P × P))
73, 6ssexi 4934 1 ~R ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1630  wex 1851  wcel 2144  Vcvv 3349  cop 4320  {copab 4844   × cxp 5247  (class class class)co 6792  Pcnp 9882   +P cpp 9884   ~R cer 9887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-om 7212  df-ni 9895  df-nq 9935  df-np 10004  df-enr 10078
This theorem is referenced by:  addsrpr  10097  mulsrpr  10098  ltsrpr  10099  0r  10102  1sr  10103  m1r  10104  addclsr  10105  mulclsr  10106  recexsrlem  10125
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