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| Mirrors > Home > MPE Home > Th. List > dfnul4 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2110, df-clel 2816. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4336. (Revised by BJ, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| dfnul4 | ⊢ ∅ = {𝑥 ∣ ⊥} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-nul 4334 | . 2 ⊢ ∅ = (V ∖ V) | |
| 2 | df-dif 3954 | . 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} | |
| 3 | pm3.24 402 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
| 4 | 3 | bifal 1556 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥) |
| 5 | 4 | abbii 2809 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥} |
| 6 | 1, 2, 5 | 3eqtri 2769 | 1 ⊢ ∅ = {𝑥 ∣ ⊥} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ⊥wfal 1552 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∖ cdif 3948 ∅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-dif 3954 df-nul 4334 |
| This theorem is referenced by: dfnul2 4336 dfnul3 4337 noel 4338 vn0 4345 eq0 4350 ab0w 4379 ab0 4380 abf 4406 eq0rdv 4407 rzal 4509 ralf0 4514 |
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