Theorem rnep 5906

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 5868	. 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 2717	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5		epel 5556	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	5	exbii 1848	. . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		neq0 4327	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
8	7	bicomi 224	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅)
9		velsn 4617	. . . . . . 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 3941	. . . . 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 3979	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
17	1, 16	eqtri 2758	1 ⊢ ran E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2713  Vcvv 3459   ∖ cdif 3923  ∅c0 4308  {csn 4601   class class class wbr 5119   E cep 5552  ran crn 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-eprel 5553  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by: (None)