Theorem reueq 3034

Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)

Assertion

Ref	Expression
reueq	⊢ (B ∈ A ↔ ∃!x ∈ A x = B)

Distinct variable groups:   x,A   x,B

Proof of Theorem reueq

Step	Hyp	Ref	Expression
1		risset 2662	. 2 ⊢ (B ∈ A ↔ ∃x ∈ A x = B)
2		moeq 3013	. . . 4 ⊢ ∃*x x = B
3		mormo 2824	. . . 4 ⊢ (∃*x x = B → ∃*x ∈ A x = B)
4	2, 3	ax-mp 5	. . 3 ⊢ ∃*x ∈ A x = B
5		reu5 2825	. . 3 ⊢ (∃!x ∈ A x = B ↔ (∃x ∈ A x = B ∧ ∃*x ∈ A x = B))
6	4, 5	mpbiran2 885	. 2 ⊢ (∃!x ∈ A x = B ↔ ∃x ∈ A x = B)
7	1, 6	bitr4i 243	1 ⊢ (B ∈ A ↔ ∃!x ∈ A x = B)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∃wrex 2616  ∃!wreu 2617  ∃*wrmo 2618

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-rex 2621  df-reu 2622  df-rmo 2623  df-v 2862

This theorem is referenced by: (None)