Theorem axextmo 2710

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 806	. . . . 5 ⊢ (((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
2	1	alanimi 1813	. . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
3		ax-ext 2706	. . . 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 1793	. 2 ⊢ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)
6		nfv 1912	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
7		axextmo.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
8	6, 7	nfbi 1901	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ 𝜑)
9	8	nfal 2322	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)
10		elequ2 2121	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
11	10	bibi1d 343	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ 𝜑) ↔ (𝑦 ∈ 𝑧 ↔ 𝜑)))
12	11	albidv 1918	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)))
13	9, 12	mo4f 2565	. 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)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  Ⅎwnf 1780  ∃*wmo 2536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538

This theorem is referenced by:  nulmo  2711