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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 4338 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
| 2 | 1 | ax-gen 1795 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅ | 
| 3 | ax-nul 5306 | . . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
| 4 | nulmo 2713 | . . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
| 5 | df-eu 2569 | . . 3 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)) | |
| 6 | 3, 4, 5 | mpbir2an 711 | . 2 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | 
| 7 | 2, 6 | pm3.2i 470 | 1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 ∃!weu 2568 ∅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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: (None) | 
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