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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 5899 | . 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
| 2 | nfab1 2907 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
| 3 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥(V ∖ {∅}) | |
| 4 | abid 2718 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥) | |
| 5 | epel 5587 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
| 6 | 5 | exbii 1848 | . . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥) |
| 7 | neq0 4352 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
| 8 | 7 | bicomi 224 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅) |
| 9 | velsn 4642 | . . . . . . 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 3966 | . . . . 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 4004 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅}) |
| 17 | 1, 16 | eqtri 2765 | 1 ⊢ ran E = (V ∖ {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 {csn 4626 class class class wbr 5143 E cep 5583 ran crn 5686 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: (None) |
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