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Mirrors > Home > MPE Home > Th. List > dfnul2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of dfnul2 4317 as of 23-Sep-2024. (Contributed by NM, 26-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfnul2OLD | ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nul 4315 | . 2 ⊢ ∅ = (V ∖ V) | |
2 | df-dif 3943 | . 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} | |
3 | pm3.24 402 | . . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) | |
4 | equid 2007 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
5 | 4 | notnoti 143 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
6 | 3, 5 | 2false 375 | . . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥) |
7 | 6 | abbii 2794 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
8 | 1, 2, 7 | 3eqtri 2756 | 1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 Vcvv 3466 ∖ cdif 3937 ∅c0 4314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-dif 3943 df-nul 4315 |
This theorem is referenced by: (None) |
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