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Mirrors > Home > MPE Home > Th. List > dfnul3 | Structured version Visualization version GIF version |
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 1557 | . . . 4 ⊢ ¬ ⊥ | |
2 | pm3.24 406 | . . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) | |
3 | 1, 2 | 2false 379 | . . 3 ⊢ (⊥ ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)) |
4 | 3 | abbii 2801 | . 2 ⊢ {𝑥 ∣ ⊥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} |
5 | dfnul4 4225 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
6 | df-rab 3060 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | |
7 | 4, 5, 6 | 3eqtr4i 2769 | 1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1543 ⊥wfal 1555 ∈ wcel 2112 {cab 2714 {crab 3055 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-rab 3060 df-dif 3856 df-nul 4224 |
This theorem is referenced by: difid 4271 kmlem3 9731 |
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