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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 5723 | . 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
2 | nfab1 2957 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
3 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑥(V ∖ {∅}) | |
4 | abid 2780 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥) | |
5 | epel 5433 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
6 | 5 | exbii 1849 | . . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥) |
7 | neq0 4259 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
8 | 7 | bicomi 227 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅) |
9 | velsn 4541 | . . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
10 | 9 | bicomi 227 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅}) |
11 | 10 | notbii 323 | . . . . 5 ⊢ (¬ 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅}) |
12 | 6, 8, 11 | 3bitri 300 | . . . 4 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅}) |
13 | velcomp 3896 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅}) | |
14 | 13 | bicomi 227 | . . . 4 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅})) |
15 | 4, 12, 14 | 3bitri 300 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅})) |
16 | 2, 3, 15 | eqri 3935 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅}) |
17 | 1, 16 | eqtri 2821 | 1 ⊢ ran E = (V ∖ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∖ cdif 3878 ∅c0 4243 {csn 4525 class class class wbr 5030 E cep 5429 ran crn 5520 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: (None) |
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