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Mirrors > Home > MPE Home > Th. List > dfnul2 | Structured version Visualization version 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 3916 | . . . 4 ⊢ ∅ = (V ∖ V) | |
2 | 1 | eleq2i 2693 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V)) |
3 | eldif 3584 | . . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)) | |
4 | eqid 2622 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
5 | pm3.24 926 | . . . . 5 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
6 | 4, 5 | 2th 254 | . . . 4 ⊢ (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)) |
7 | 6 | con2bii 347 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥) |
8 | 2, 3, 7 | 3bitri 286 | . 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥) |
9 | 8 | abbi2i 2738 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 Vcvv 3200 ∖ cdif 3571 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: dfnul3 3918 rab0OLD 3956 iotanul 5866 avril1 27319 |
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