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| Mirrors > Home > MPE Home > Th. List > dfnul2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2141, ax-11 2157, and ax-12 2177. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| dfnul2 | ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfnul4 4335 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
| 2 | equid 2011 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
| 3 | 2 | notnoti 143 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
| 4 | 3 | bifal 1556 | . . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥) |
| 5 | 4 | abbii 2809 | . 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥} |
| 6 | 1, 5 | eqtr4i 2768 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ⊥wfal 1552 {cab 2714 ∅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: ab0orv 4383 iotanul 6539 avril1 30482 |
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