Theorem enrex 11136

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 11055	. . . 4 ⊢ P ∈ V
2	1, 1	xpex 7788	. . 3 ⊢ (P × P) ∈ V
3	2, 2	xpex 7788	. 2 ⊢ ((P × P) × (P × P)) ∈ V
4		df-enr 11124	. . 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 5792	. . 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 4043	. 2 ⊢ ~R ⊆ ((P × P) × (P × P))
7	3, 6	ssexi 5340	1 ⊢ ~R ∈ V

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654  {copab 5228   × cxp 5698  (class class class)co 7448  Pcnp 10928   +_P cpp 10930   ~_R cer 10933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-om 7904  df-ni 10941  df-nq 10981  df-np 11050  df-enr 11124

This theorem is referenced by:  addsrpr  11144  mulsrpr  11145  ltsrpr  11146  0r  11149  1sr  11150  m1r  11151  addclsr  11152  mulclsr  11153  recexsrlem  11172