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Theorem rnep 5937
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 5899 . 2 ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2 nfab1 2907 . . 3 𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3 nfcv 2905 . . 3 𝑥(V ∖ {∅})
4 abid 2718 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5 epel 5587 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
65exbii 1848 . . . . 5 (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦𝑥)
7 neq0 4352 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
87bicomi 224 . . . . 5 (∃𝑦 𝑦𝑥 ↔ ¬ 𝑥 = ∅)
9 velsn 4642 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
109bicomi 224 . . . . . 6 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
1110notbii 320 . . . . 5 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
126, 8, 113bitri 297 . . . 4 (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13 velcomp 3966 . . . . 5 (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
1413bicomi 224 . . . 4 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
154, 12, 143bitri 297 . . 3 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
162, 3, 15eqri 4004 . 2 {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
171, 16eqtri 2765 1 ran E = (V ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wex 1779  wcel 2108  {cab 2714  Vcvv 3480  cdif 3948  c0 4333  {csn 4626   class class class wbr 5143   E cep 5583  ran crn 5686
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by: (None)
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