Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 4296 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | ax-gen 1796 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅ |
3 | ax-nul 5210 | . . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
4 | nulmo 2798 | . . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
5 | df-eu 2654 | . . 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 473 | 1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∀wal 1535 ∃wex 1780 ∈ wcel 2114 ∃*wmo 2620 ∃!weu 2653 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-dif 3939 df-nul 4292 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |