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Theorem rnep 5790
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 5752 . 2 ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2 nfab1 2977 . . 3 𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3 nfcv 2975 . . 3 𝑥(V ∖ {∅})
4 abid 2801 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5 epel 5462 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
65exbii 1842 . . . . 5 (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦𝑥)
7 neq0 4307 . . . . . 6 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
87bicomi 226 . . . . 5 (∃𝑦 𝑦𝑥 ↔ ¬ 𝑥 = ∅)
9 velsn 4575 . . . . . . 7 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
109bicomi 226 . . . . . 6 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
1110notbii 322 . . . . 5 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
126, 8, 113bitri 299 . . . 4 (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13 velcomp 3949 . . . . 5 (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
1413bicomi 226 . . . 4 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
154, 12, 143bitri 299 . . 3 (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
162, 3, 15eqri 3985 . 2 {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
171, 16eqtri 2842 1 ran E = (V ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1531  wex 1774  wcel 2108  {cab 2797  Vcvv 3493  cdif 3931  c0 4289  {csn 4559   class class class wbr 5057   E cep 5457  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-eprel 5458  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by: (None)
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