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Mirrors > Home > ILE Home > Th. List > dfnul2 | GIF version |
Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) |
Ref | Expression |
---|---|
dfnul2 | ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nul 3269 | . . . 4 ⊢ ∅ = (V ∖ V) | |
2 | 1 | eleq2i 2149 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V)) |
3 | eldif 2992 | . . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)) | |
4 | pm3.24 660 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
5 | eqid 2083 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
6 | 5 | notnoti 607 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
7 | 4, 6 | 2false 650 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥) |
8 | 2, 3, 7 | 3bitri 204 | . 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥) |
9 | 8 | abbi2i 2197 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 = wceq 1285 ∈ wcel 1434 {cab 2069 Vcvv 2611 ∖ cdif 2980 ∅c0 3268 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2613 df-dif 2985 df-nul 3269 |
This theorem is referenced by: dfnul3 3271 rab0 3290 iotanul 4933 |
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