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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 2139, ax-11 2155, and ax-12 2175. (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 4341 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
2 | equid 2009 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
3 | 2 | notnoti 143 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
4 | 3 | bifal 1553 | . . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥) |
5 | 4 | abbii 2807 | . 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥} |
6 | 1, 5 | eqtr4i 2766 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ⊥wfal 1549 {cab 2712 ∅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-dif 3966 df-nul 4340 |
This theorem is referenced by: ab0orv 4389 iotanul 6541 avril1 30492 |
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