Theorem enrex 11107

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 11026	. . . 4 ⊢ P ∈ V
2	1, 1	xpex 7773	. . 3 ⊢ (P × P) ∈ V
3	2, 2	xpex 7773	. 2 ⊢ ((P × P) × (P × P)) ∈ V
4		df-enr 11095	. . 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 5778	. . 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 4030	. 2 ⊢ ~R ⊆ ((P × P) × (P × P))
7	3, 6	ssexi 5322	1 ⊢ ~R ∈ V

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632  {copab 5205   × cxp 5683  (class class class)co 7431  Pcnp 10899   +_P cpp 10901   ~_R cer 10904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888  df-ni 10912  df-nq 10952  df-np 11021  df-enr 11095

This theorem is referenced by:  addsrpr  11115  mulsrpr  11116  ltsrpr  11117  0r  11120  1sr  11121  m1r  11122  addclsr  11123  mulclsr  11124  recexsrlem  11143