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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 2077, ax-11 2091, and ax-12 2104. (Revised by Steven Nguyen, 3-May-2023.) |
Ref | Expression |
---|---|
dfnul2 | ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nul 4178 | . 2 ⊢ ∅ = (V ∖ V) | |
2 | df-dif 3831 | . 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} | |
3 | pm3.24 394 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
4 | equid 1968 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
5 | 4 | notnoti 140 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
6 | 3, 5 | 2false 368 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥) |
7 | 6 | abbii 2841 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
8 | 1, 2, 7 | 3eqtri 2803 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 387 = wceq 1507 ∈ wcel 2048 {cab 2755 Vcvv 3412 ∖ cdif 3825 ∅c0 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-9 2057 ax-ext 2747 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-sb 2014 df-clab 2756 df-cleq 2768 df-dif 3831 df-nul 4178 |
This theorem is referenced by: dfnul3 4181 iotanul 6165 avril1 28013 |
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