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Mirrors > Home > MPE Home > Th. List > rnep | Structured version Visualization version GIF version |
Description: The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.) |
Ref | Expression |
---|---|
rnep | ⊢ ran E = (V ∖ {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 5886 | . 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
2 | nfab1 2905 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
3 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑥(V ∖ {∅}) | |
4 | abid 2713 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥) | |
5 | epel 5582 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
6 | 5 | exbii 1850 | . . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥) |
7 | neq0 4344 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
8 | 7 | bicomi 223 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅) |
9 | velsn 4643 | . . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
10 | 9 | bicomi 223 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅}) |
11 | 10 | notbii 319 | . . . . 5 ⊢ (¬ 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅}) |
12 | 6, 8, 11 | 3bitri 296 | . . . 4 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅}) |
13 | velcomp 3962 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅}) | |
14 | 13 | bicomi 223 | . . . 4 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅})) |
15 | 4, 12, 14 | 3bitri 296 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅})) |
16 | 2, 3, 15 | eqri 4001 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅}) |
17 | 1, 16 | eqtri 2760 | 1 ⊢ ran E = (V ∖ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 Vcvv 3474 ∖ cdif 3944 ∅c0 4321 {csn 4627 class class class wbr 5147 E cep 5578 ran crn 5676 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-eprel 5579 df-cnv 5683 df-dm 5685 df-rn 5686 |
This theorem is referenced by: (None) |
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