MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  enrex Structured version   Visualization version   GIF version

Theorem enrex 10140
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 10060 . . . 4 P ∈ V
21, 1xpex 7159 . . 3 (P × P) ∈ V
32, 2xpex 7159 . 2 ((P × P) × (P × P)) ∈ V
4 df-enr 10129 . . 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 5362 . . 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 3794 . 2 ~R ⊆ ((P × P) × (P × P))
73, 6ssexi 4963 1 ~R ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wex 1874  wcel 2155  Vcvv 3349  cop 4339  {copab 4870   × cxp 5274  (class class class)co 6841  Pcnp 9933   +P cpp 9935   ~R cer 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-tr 4911  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-om 7263  df-ni 9946  df-nq 9986  df-np 10055  df-enr 10129
This theorem is referenced by:  addsrpr  10148  mulsrpr  10149  ltsrpr  10150  0r  10153  1sr  10154  m1r  10155  addclsr  10156  mulclsr  10157  recexsrlem  10176
  Copyright terms: Public domain W3C validator