Theorem eusn 1913

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton.

Assertion

Ref	Expression
eusn	⊢ (∃!xφ ↔ ∃x{x∣φ} = {x})

Proof of Theorem eusn

Step	Hyp	Ref	Expression
1		cleqabr 1175	. . . 4 ⊢ ({x∣φ} = {y} ↔ ∀x(φ ↔ x ∈ {y}))
2		elsn 1820	. . . . . 6 ⊢ (x ∈ {y} ↔ x = y)
3	2	bibi2i 460	. . . . 5 ⊢ ((φ ↔ x ∈ {y}) ↔ (φ ↔ x = y))
4	3	bial 695	. . . 4 ⊢ (∀x(φ ↔ x ∈ {y}) ↔ ∀x(φ ↔ x = y))
5	1, 4	bitr 151	. . 3 ⊢ ({x∣φ} = {y} ↔ ∀x(φ ↔ x = y))
6	5	biex 733	. 2 ⊢ (∃y{x∣φ} = {y} ↔ ∃y∀x(φ ↔ x = y))
7		ax-17 925	. . 3 ⊢ ({x∣φ} = {x} → ∀y{x∣φ} = {x})
8		hbab1 1095	. . . 4 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
9		ax-17 925	. . . 4 ⊢ (z ∈ {y} → ∀x z ∈ {y})
10	8, 9	hbeq 1171	. . 3 ⊢ ({x∣φ} = {y} → ∀x{x∣φ} = {y})
11		sneq 1816	. . . 4 ⊢ (x = y → {x} = {y})
12	11	cleq2d 1112	. . 3 ⊢ (x = y → ({x∣φ} = {x} ↔ {x∣φ} = {y}))
13	7, 10, 12	cbvex 849	. 2 ⊢ (∃x{x∣φ} = {x} ↔ ∃y{x∣φ} = {y})
14		df-eu 1009	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
15	6, 13, 14	3bitr4r 159	1 ⊢ (∃!xφ ↔ ∃x{x∣φ} = {x})

Syntax hints:   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007  {cab 1090   = wceq 1091   ∈ wcel 1092  {csn 1808

This theorem is referenced by:  euuni 1954  reucl 1957  mapsn 3269

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-sn 1811

metamath.org