Theorem axextmo 2739

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.)

Hypothesis

Ref	Expression
axextmo.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
axextmo	⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axextmo

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biantr 817	. . . . 5 ⊢ (((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
2	1	alanimi 1846	. . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
3		ax-ext 2735	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑧)
4	2, 3	syl 18	. . 3 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)
5	4	gen2 1826	. 2 ⊢ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)
6		nfv 1944	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
7		axextmo.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
8	6, 7	nfbi 1933	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ 𝜑)
9	8	nfal 2356	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)
10		elequ2 2158	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
11	10	bibi1d 346	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ 𝜑) ↔ (𝑦 ∈ 𝑧 ↔ 𝜑)))
12	11	albidv 1950	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)))
13	9, 12	mo4f 2595	. 2 ⊢ (∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧))
14	5, 13	mpbir 234	1 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  Ⅎwnf 1813  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567

This theorem is referenced by:  nulmo  2740