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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 2108, df-clel 2816. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4259. (Revised by BJ, 23-Sep-2024.) |
Ref | Expression |
---|---|
dfnul4 | ⊢ ∅ = {𝑥 ∣ ⊥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nul 4257 | . 2 ⊢ ∅ = (V ∖ V) | |
2 | df-dif 3890 | . 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} | |
3 | pm3.24 403 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
4 | 3 | bifal 1555 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥) |
5 | 4 | abbii 2808 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥} |
6 | 1, 2, 5 | 3eqtri 2770 | 1 ⊢ ∅ = {𝑥 ∣ ⊥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1539 ⊥wfal 1551 ∈ wcel 2106 {cab 2715 Vcvv 3432 ∖ cdif 3884 ∅c0 4256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-dif 3890 df-nul 4257 |
This theorem is referenced by: dfnul2 4259 dfnul3 4260 noel 4264 vn0 4272 eq0 4277 ab0w 4307 ab0 4308 ab0OLD 4309 abf 4336 eq0rdv 4338 rzal 4439 ralf0 4444 |
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