Theorem eueq 1427

Description: Equality has existential uniqueness.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem eueq

Step	Hyp	Ref	Expression
1		cleq2 1110	. . . . 5 ⊢ (y = A → (x = y ↔ x = A))
2	1	biimparc 327	. . . 4 ⊢ ((x = A ∧ y = A) → x = y)
3	2	gen2 681	. . 3 ⊢ ∀x∀y((x = A ∧ y = A) → x = y)
4	3	biantru 543	. 2 ⊢ (∃x x = A ↔ (∃x x = A ∧ ∀x∀y((x = A ∧ y = A) → x = y)))
5		isset 1351	. 2 ⊢ (A ∈ V ↔ ∃x x = A)
6		cleq1 1107	. . 3 ⊢ (x = y → (x = A ↔ y = A))
7	6	eu4 1036	. 2 ⊢ (∃!x x = A ↔ (∃x x = A ∧ ∀x∀y((x = A ∧ y = A) → x = y)))
8	4, 5, 7	3bitr4 158	1 ⊢ (A ∈ V ↔ ∃!x x = A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  eueq1 1428  moeq 1431  reuhyp 1581  euuni 1954  fvopab2 2878  fopab2 2891  en2d 3303

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-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

metamath.org