Theorem enrex 11070

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 10989	. . . 4 ⊢ P ∈ V
2	1, 1	xpex 7761	. . 3 ⊢ (P × P) ∈ V
3	2, 2	xpex 7761	. 2 ⊢ ((P × P) × (P × P)) ∈ V
4		df-enr 11058	. . 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 5758	. . 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 3986	. 2 ⊢ ~R ⊆ ((P × P) × (P × P))
7	3, 6	ssexi 5298	1 ⊢ ~R ∈ V

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3458  ⟨cop 4600  {copab 5178   × cxp 5664  (class class class)co 7423  Pcnp 10862   +_P cpp 10864   ~_R cer 10867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-om 7872  df-ni 10875  df-nq 10915  df-np 10984  df-enr 11058

This theorem is used by:  addsrpr  11078  mulsrpr  11079  ltsrpr  11080  0r  11083  1sr  11084  m1r  11085  addclsr  11086  mulclsr  11087  recexsrlem  11106