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Mirrors > Home > ILE Home > Th. List > zfnuleu | GIF version |
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2174 to strengthen the hypothesis in the form of axnul 4143). (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 1375 | . . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥)) | |
3 | 2 | albii 1481 | . . . . 5 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
4 | 3 | exbii 1616 | . . . 4 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
5 | 1, 4 | mpbi 145 | . . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
6 | nfv 1539 | . . . 4 ⊢ Ⅎ𝑥⊥ | |
7 | 6 | bm1.1 2174 | . . 3 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
9 | 3 | eubii 2047 | . 2 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
10 | 8, 9 | mpbir 146 | 1 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 ∀wal 1362 ⊥wfal 1369 ∃wex 1503 ∃!weu 2038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 |
This theorem is referenced by: (None) |
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