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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 2073 to strengthen the hypothesis in the form of axnul 3964). (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 1300 | . . . . . 6 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥)) | |
3 | 2 | albii 1404 | . . . . 5 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
4 | 3 | exbii 1541 | . . . 4 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
5 | 1, 4 | mpbi 143 | . . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
6 | nfv 1466 | . . . 4 ⊢ Ⅎ𝑥⊥ | |
7 | 6 | bm1.1 2073 | . . 3 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
8 | 5, 7 | ax-mp 7 | . 2 ⊢ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
9 | 3 | eubii 1957 | . 2 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
10 | 8, 9 | mpbir 144 | 1 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∀wal 1287 ⊥wfal 1294 ∃wex 1426 ∃!weu 1948 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 |
This theorem is referenced by: (None) |
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