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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 805 | . . . . 5 ⊢ (((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)) | |
2 | 1 | alanimi 1819 | . . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)) |
3 | ax-ext 2704 | . . . 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 1799 | . 2 ⊢ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧) |
6 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
7 | axextmo.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
8 | 6, 7 | nfbi 1907 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ 𝜑) |
9 | 8 | nfal 2317 | . . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑) |
10 | elequ2 2122 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)) | |
11 | 10 | bibi1d 344 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ 𝜑) ↔ (𝑦 ∈ 𝑧 ↔ 𝜑))) |
12 | 11 | albidv 1924 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑))) |
13 | 9, 12 | mo4f 2562 | . 2 ⊢ (∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)) |
14 | 5, 13 | mpbir 230 | 1 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 Ⅎwnf 1786 ∃*wmo 2533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 |
This theorem is referenced by: nulmo 2709 |
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