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Mirrors > Home > MPE Home > Th. List > zfnuleu | Structured version Visualization version GIF version |
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2595 to strengthen the hypothesis in the form of axnul 4716). (Contributed by NM, 22-Dec-2007.) |
Ref | Expression |
---|---|
zfnuleu.1 | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Ref | Expression |
---|---|
zfnuleu | ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfnuleu.1 | . . . 4 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | |
2 | nbfal 1486 | . . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥)) | |
3 | 2 | albii 1737 | . . . . 5 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
4 | 3 | exbii 1764 | . . . 4 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
5 | 1, 4 | mpbi 219 | . . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
6 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑥⊥ | |
7 | 6 | bm1.1 2595 | . . 3 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
9 | 3 | eubii 2480 | . 2 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
10 | 8, 9 | mpbir 220 | 1 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 ⊥wfal 1480 ∃wex 1695 ∃!weu 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 |
This theorem is referenced by: (None) |
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