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Theorem rnep 5917
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 5879 . 2 ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2 nfab1 2897 . . 3 𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3 nfcv 2895 . . 3 𝑥(V ∖ {∅})
4 abid 2705 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5 epel 5574 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
65exbii 1842 . . . . 5 (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦𝑥)
7 neq0 4338 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
87bicomi 223 . . . . 5 (∃𝑦 𝑦𝑥 ↔ ¬ 𝑥 = ∅)
9 velsn 4637 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
109bicomi 223 . . . . . 6 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
1110notbii 320 . . . . 5 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
126, 8, 113bitri 297 . . . 4 (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13 velcomp 3956 . . . . 5 (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
1413bicomi 223 . . . 4 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
154, 12, 143bitri 297 . . 3 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
162, 3, 15eqri 3995 . 2 {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
171, 16eqtri 2752 1 ran E = (V ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wex 1773  wcel 2098  {cab 2701  Vcvv 3466  cdif 3938  c0 4315  {csn 4621   class class class wbr 5139   E cep 5570  ran crn 5668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-cnv 5675  df-dm 5677  df-rn 5678
This theorem is referenced by: (None)
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