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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 2109, df-clel 2811. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4323. (Revised by BJ, 23-Sep-2024.) |
Ref | Expression |
---|---|
dfnul4 | ⊢ ∅ = {𝑥 ∣ ⊥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nul 4321 | . 2 ⊢ ∅ = (V ∖ V) | |
2 | df-dif 3949 | . 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} | |
3 | pm3.24 404 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
4 | 3 | bifal 1558 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥) |
5 | 4 | abbii 2803 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥} |
6 | 1, 2, 5 | 3eqtri 2765 | 1 ⊢ ∅ = {𝑥 ∣ ⊥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ⊥wfal 1554 ∈ wcel 2107 {cab 2710 Vcvv 3475 ∖ cdif 3943 ∅c0 4320 |
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 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-dif 3949 df-nul 4321 |
This theorem is referenced by: dfnul2 4323 dfnul3 4324 noel 4328 vn0 4336 eq0 4341 ab0w 4371 ab0 4372 ab0OLD 4373 abf 4400 eq0rdv 4402 rzal 4506 ralf0 4511 |
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