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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 5845 | . 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 5541 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
6 | 5 | exbii 1851 | . . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥) |
7 | neq0 4306 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
8 | 7 | bicomi 223 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅) |
9 | velsn 4603 | . . . . . . 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 3926 | . . . . 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 3965 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅}) |
17 | 1, 16 | eqtri 2761 | 1 ⊢ ran E = (V ∖ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 Vcvv 3444 ∖ cdif 3908 ∅c0 4283 {csn 4587 class class class wbr 5106 E cep 5537 ran crn 5635 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-eprel 5538 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: (None) |
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