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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 4291 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | ax-gen 1798 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅ |
3 | ax-nul 5264 | . . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
4 | nulmo 2713 | . . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
5 | df-eu 2568 | . . 3 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)) | |
6 | 3, 4, 5 | mpbir2an 710 | . 2 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
7 | 2, 6 | pm3.2i 472 | 1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∀wal 1540 ∃wex 1782 ∈ wcel 2107 ∃*wmo 2537 ∃!weu 2567 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-dif 3914 df-nul 4284 |
This theorem is referenced by: (None) |
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