Theorem rnep 5921

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 5883	. 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2		nfab1 2906	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3		nfcv 2904	. . 3 ⊢ Ⅎ𝑥(V ∖ {∅})
4		abid 2714	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5		epel 5579	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	5	exbii 1851	. . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		neq0 4343	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
8	7	bicomi 223	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅)
9		velsn 4640	. . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
10	9	bicomi 223	. . . . . 6 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
11	10	notbii 320	. . . . 5 ⊢ (¬ 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
12	6, 8, 11	3bitri 297	. . . 4 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13		velcomp 3961	. . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
14	13	bicomi 223	. . . 4 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
15	4, 12, 14	3bitri 297	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
16	2, 3, 15	eqri 4000	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
17	1, 16	eqtri 2761	1 ⊢ ran E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  Vcvv 3475   ∖ cdif 3943  ∅c0 4320  {csn 4624   class class class wbr 5144   E cep 5575  ran crn 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-eprel 5576  df-cnv 5680  df-dm 5682  df-rn 5683

This theorem is referenced by: (None)