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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 3438 | . . . 4 ⊢ ∅ = (V ∖ V) | |
2 | 1 | eleq2i 2256 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V)) |
3 | eldif 3153 | . . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)) | |
4 | pm3.24 694 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
5 | eqid 2189 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
6 | 5 | notnoti 646 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
7 | 4, 6 | 2false 702 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥) |
8 | 2, 3, 7 | 3bitri 206 | . 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥) |
9 | 8 | abbi2i 2304 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 = wceq 1364 ∈ wcel 2160 {cab 2175 Vcvv 2752 ∖ cdif 3141 ∅c0 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-nul 3438 |
This theorem is referenced by: dfnul3 3440 rab0 3466 iotanul 5211 |
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