Theorem rnep 5882

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 5843	. 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2		nfab1 2900	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3		nfcv 2898	. . 3 ⊢ Ⅎ𝑥(V ∖ {∅})
4		abid 2718	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5		epel 5534	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	5	exbii 1850	. . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		neq0 4292	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
8	7	bicomi 224	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅)
9		velsn 4583	. . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
10	9	bicomi 224	. . . . . 6 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
11	10	notbii 320	. . . . 5 ⊢ (¬ 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅})
12	6, 8, 11	3bitri 297	. . . 4 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅})
13		velcomp 3904	. . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅})
14	13	bicomi 224	. . . 4 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅}))
15	4, 12, 14	3bitri 297	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅}))
16	2, 3, 15	eqri 3942	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
17	1, 16	eqtri 2759	1 ⊢ ran E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  Vcvv 3429   ∖ cdif 3886  ∅c0 4273  {csn 4567   class class class wbr 5085   E cep 5530  ran crn 5632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-eprel 5531  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by: (None)