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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 5759 | . 2 ⊢ ran E = {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
2 | nfab1 2979 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∃𝑦 𝑦 E 𝑥} | |
3 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑥(V ∖ {∅}) | |
4 | abid 2803 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ ∃𝑦 𝑦 E 𝑥) | |
5 | epel 5469 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
6 | 5 | exbii 1848 | . . . . 5 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥) |
7 | neq0 4309 | . . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥) | |
8 | 7 | bicomi 226 | . . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ 𝑥 = ∅) |
9 | velsn 4583 | . . . . . . 7 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
10 | 9 | bicomi 226 | . . . . . 6 ⊢ (𝑥 = ∅ ↔ 𝑥 ∈ {∅}) |
11 | 10 | notbii 322 | . . . . 5 ⊢ (¬ 𝑥 = ∅ ↔ ¬ 𝑥 ∈ {∅}) |
12 | 6, 8, 11 | 3bitri 299 | . . . 4 ⊢ (∃𝑦 𝑦 E 𝑥 ↔ ¬ 𝑥 ∈ {∅}) |
13 | velcomp 3951 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ ¬ 𝑥 ∈ {∅}) | |
14 | 13 | bicomi 226 | . . . 4 ⊢ (¬ 𝑥 ∈ {∅} ↔ 𝑥 ∈ (V ∖ {∅})) |
15 | 4, 12, 14 | 3bitri 299 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} ↔ 𝑥 ∈ (V ∖ {∅})) |
16 | 2, 3, 15 | eqri 3987 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 E 𝑥} = (V ∖ {∅}) |
17 | 1, 16 | eqtri 2844 | 1 ⊢ ran E = (V ∖ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 {csn 4567 class class class wbr 5066 E cep 5464 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: (None) |
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