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Mirrors > Home > MPE Home > Th. List > Mathboxes > eu0 | Structured version Visualization version GIF version |
Description: There is only one empty set. (Contributed by RP, 1-Oct-2023.) |
Ref | Expression |
---|---|
eu0 | ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4329 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | ax-gen 1797 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅ |
3 | ax-nul 5305 | . . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
4 | nulmo 2708 | . . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
5 | df-eu 2563 | . . 3 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)) | |
6 | 3, 4, 5 | mpbir2an 709 | . 2 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
7 | 2, 6 | pm3.2i 471 | 1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∀wal 1539 ∃wex 1781 ∈ wcel 2106 ∃*wmo 2532 ∃!weu 2562 ∅c0 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-dif 3950 df-nul 4322 |
This theorem is referenced by: (None) |
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