Theorem rnep 5939

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 5901	. 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
2		nfab1 2904	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥}
3		nfcv 2902	. . 3 ⊢ Ⅎ𝑥(V ∖ {∅})
4		abid 2715	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥)
5		epel 5591	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	5	exbii 1844	. . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		neq0 4357	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
8	7	bicomi 224	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅)
9		velsn 4646	. . . . . . 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 3977	. . . . 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 4015	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅})
17	1, 16	eqtri 2762	1 ⊢ ran E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1536  ∃wex 1775   ∈ wcel 2105  {cab 2711  Vcvv 3477   ∖ cdif 3959  ∅c0 4338  {csn 4630   class class class wbr 5147   E cep 5587  ran crn 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-eprel 5588  df-cnv 5696  df-dm 5698  df-rn 5699

This theorem is referenced by: (None)