| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > axextmo | Structured version Visualization version GIF version | ||
| Description: There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) |
| Ref | Expression |
|---|---|
| axextmo.1 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| axextmo | ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biantr 806 | . . . . 5 ⊢ (((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)) | |
| 2 | 1 | alanimi 1816 | . . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)) |
| 3 | ax-ext 2708 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑧) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧) |
| 5 | 4 | gen2 1796 | . 2 ⊢ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧) |
| 6 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
| 7 | axextmo.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
| 8 | 6, 7 | nfbi 1903 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ 𝜑) |
| 9 | 8 | nfal 2323 | . . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑) |
| 10 | elequ2 2123 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)) | |
| 11 | 10 | bibi1d 343 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ 𝜑) ↔ (𝑦 ∈ 𝑧 ↔ 𝜑))) |
| 12 | 11 | albidv 1920 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑))) |
| 13 | 9, 12 | mo4f 2567 | . 2 ⊢ (∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)) |
| 14 | 5, 13 | mpbir 231 | 1 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 Ⅎwnf 1783 ∃*wmo 2538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 |
| This theorem is referenced by: nulmo 2713 |
| Copyright terms: Public domain | W3C validator |