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Theorem rnep 5764
Description: The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.)
Assertion
Ref Expression
rnep ran E = (V ∖ {∅})

Proof of Theorem rnep
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 5725 . 2 ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2 nfab1 2901 . . 3 𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3 nfcv 2899 . . 3 𝑥(V ∖ {∅})
4 abid 2720 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5 epel 5433 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
65exbii 1854 . . . . 5 (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦𝑥)
7 neq0 4232 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
87bicomi 227 . . . . 5 (∃𝑦 𝑦𝑥 ↔ ¬ 𝑥 = ∅)
9 velsn 4529 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
109bicomi 227 . . . . . 6 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
1110notbii 323 . . . . 5 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
126, 8, 113bitri 300 . . . 4 (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13 velcomp 3856 . . . . 5 (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
1413bicomi 227 . . . 4 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
154, 12, 143bitri 300 . . 3 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
162, 3, 15eqri 3895 . 2 {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
171, 16eqtri 2761 1 ran E = (V ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wex 1786  wcel 2113  {cab 2716  Vcvv 3397  cdif 3838  c0 4209  {csn 4513   class class class wbr 5027   E cep 5429  ran crn 5520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-v 3399  df-dif 3844  df-un 3846  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-eprel 5430  df-cnv 5527  df-dm 5529  df-rn 5530
This theorem is referenced by: (None)
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