Theorem enrex 10990

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.

Step	Hyp	Ref	Expression
1		npex 10909	. . . 4 ⊢ P ∈ V
2	1, 1	xpex 7708	. . 3 ⊢ (P × P) ∈ V
3	2, 2	xpex 7708	. 2 ⊢ ((P × P) × (P × P)) ∈ V
4		df-enr 10978	. . 3 ⊢ ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
5		opabssxp 5724	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} ⊆ ((P × P) × (P × P))
6	4, 5	eqsstri 3982	. 2 ⊢ ~R ⊆ ((P × P) × (P × P))
7	3, 6	ssexi 5269	1 ⊢ ~R ∈ V

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588  {copab 5162   × cxp 5630  (class class class)co 7368  Pcnp 10782   +_P cpp 10784   ~_R cer 10787

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-om 7819  df-ni 10795  df-nq 10835  df-np 10904  df-enr 10978

This theorem is referenced by:  addsrpr  10998  mulsrpr  10999  ltsrpr  11000  0r  11003  1sr  11004  m1r  11005  addclsr  11006  mulclsr  11007  recexsrlem  11026