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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 1551 | . . . 4 ⊢ ¬ ⊥ | |
2 | pm3.24 402 | . . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) | |
3 | 1, 2 | 2false 375 | . . 3 ⊢ (⊥ ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)) |
4 | 3 | abbii 2807 | . 2 ⊢ {𝑥 ∣ ⊥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} |
5 | dfnul4 4341 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
6 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | |
7 | 4, 5, 6 | 3eqtr4i 2773 | 1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ⊥wfal 1549 ∈ wcel 2106 {cab 2712 {crab 3433 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-rab 3434 df-dif 3966 df-nul 4340 |
This theorem is referenced by: difid 4382 kmlem3 10191 |
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