Theorem enrex 8048

Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)

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 7784	. . . 4 ⊢ P ∈ V
2	1, 1	xpex 4865	. . 3 ⊢ (P × P) ∈ V
3	2, 2	xpex 4865	. 2 ⊢ ((P × P) × (P × P)) ∈ V
4		df-enr 8037	. . 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 4823	. . 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 3269	. 2 ⊢ ~R ⊆ ((P × P) × (P × P))
7	3, 6	ssexi 4247	1 ⊢ ~R ∈ V

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2812  ⟨cop 3691  {copab 4169   × cxp 4746  (class class class)co 6049  Pcnp 7602   +_P cpp 7604   ~_R cer 7607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-qs 6772  df-ni 7615  df-nqqs 7659  df-inp 7777  df-enr 8037

This theorem is referenced by:  addsrpr  8056  mulsrpr  8057  ltsrprg  8058  0r  8061  1sr  8062  m1r  8063  addclsr  8064  mulclsr  8065  recexgt0sr  8084  prsrcl  8095  ltpsrprg  8114  mappsrprg  8115  suplocsrlemb  8117  pitonnlem2  8158  pitonn  8159  pitore  8161  recnnre  8162