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| Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| dfnul3 | ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fal 1554 | . . . 4 ⊢ ¬ ⊥ | |
| 2 | pm3.24 402 | . . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) | |
| 3 | 1, 2 | 2false 375 | . . 3 ⊢ (⊥ ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)) | 
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑥 ∣ ⊥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | 
| 5 | dfnul4 4335 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
| 6 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | |
| 7 | 4, 5, 6 | 3eqtr4i 2775 | 1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ⊥wfal 1552 ∈ wcel 2108 {cab 2714 {crab 3436 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rab 3437 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: difid 4376 kmlem3 10193 | 
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