MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnep Structured version   Visualization version   GIF version

Theorem rnep 5924
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 5886 . 2 ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2 nfab1 2905 . . 3 𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3 nfcv 2903 . . 3 𝑥(V ∖ {∅})
4 abid 2713 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5 epel 5582 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
65exbii 1850 . . . . 5 (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦𝑥)
7 neq0 4344 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
87bicomi 223 . . . . 5 (∃𝑦 𝑦𝑥 ↔ ¬ 𝑥 = ∅)
9 velsn 4643 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
109bicomi 223 . . . . . 6 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
1110notbii 319 . . . . 5 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
126, 8, 113bitri 296 . . . 4 (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13 velcomp 3962 . . . . 5 (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
1413bicomi 223 . . . 4 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
154, 12, 143bitri 296 . . 3 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
162, 3, 15eqri 4001 . 2 {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
171, 16eqtri 2760 1 ran E = (V ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wex 1781  wcel 2106  {cab 2709  Vcvv 3474  cdif 3944  c0 4321  {csn 4627   class class class wbr 5147   E cep 5578  ran crn 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-eprel 5579  df-cnv 5683  df-dm 5685  df-rn 5686
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator