Theorem rnep 5916

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.

Step	Hyp	Ref	Expression
1		dfrn2 5877	. 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2		nfab1 2928	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3		nfcv 2926	. . 3 ⊢ Ⅎ𝑥(V ∖ {∅})
4		abid 2745	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5		epel 5558	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	5	exbii 1881	. . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		neq0 4302	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
8	7	bicomi 227	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅)
9		velsn 4600	. . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
10	9	bicomi 227	. . . . . 6 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
11	10	notbii 323	. . . . 5 ⊢ (¬ 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
12	6, 8, 11	3bitri 300	. . . 4 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13		velcomp 4057	. . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
14	13	bicomi 227	. . . 4 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
15	4, 12, 14	3bitri 300	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
16	2, 3, 15	eqri 3933	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
17	1, 16	eqtri 2786	1 ⊢ ran E = (V ∖ {∅})

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  {csn 4588   class class class wbr 5106   E cep 5554  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-eprel 5555  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by: (None)