Theorem eueq 3009

Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eueq	⊢ (A ∈ V ↔ ∃!x x = A)

Distinct variable group:   x,A

Proof of Theorem eueq

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2372	. . . 4 ⊢ ((x = A ∧ y = A) → x = y)
2	1	gen2 1547	. . 3 ⊢ ∀x∀y((x = A ∧ y = A) → x = y)
3	2	biantru 491	. 2 ⊢ (∃x x = A ↔ (∃x x = A ∧ ∀x∀y((x = A ∧ y = A) → x = y)))
4		isset 2864	. 2 ⊢ (A ∈ V ↔ ∃x x = A)
5		eqeq1 2359	. . 3 ⊢ (x = y → (x = A ↔ y = A))
6	5	eu4 2243	. 2 ⊢ (∃!x x = A ↔ (∃x x = A ∧ ∀x∀y((x = A ∧ y = A) → x = y)))
7	3, 4, 6	3bitr4i 268	1 ⊢ (A ∈ V ↔ ∃!x x = A)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  eueq1  3010  moeq  3013  fnopab2g  5207