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Mirrors > Home > MPE Home > Th. List > dfnul2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of dfnul2 4174 as of 3-May-2023. Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dfnul2OLD | ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nul 4173 | . . . 4 ⊢ ∅ = (V ∖ V) | |
2 | 1 | eleq2i 2851 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V)) |
3 | eldif 3833 | . . 3 ⊢ (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)) | |
4 | eqid 2772 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
5 | pm3.24 394 | . . . . 5 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
6 | 4, 5 | 2th 256 | . . . 4 ⊢ (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)) |
7 | 6 | con2bii 350 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥) |
8 | 2, 3, 7 | 3bitri 289 | . 2 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥) |
9 | 8 | abbi2i 2899 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {cab 2752 Vcvv 3409 ∖ cdif 3820 ∅c0 4172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-dif 3826 df-nul 4173 |
This theorem is referenced by: (None) |
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