Theorem reueq 3733

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 3229	. 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵)
2		moeq 3703	. . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵
3		mormo 3380	. . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)
4	2, 3	ax-mp 5	. . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵
5		reu5 3377	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵))
6	4, 5	mpbiran2 707	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵)
7	1, 6	bitr4i 278	1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1540   ∈ wcel 2105  ∃*wmo 2531  ∃wrex 3069  ∃!wreu 3373  ∃*wrmo 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-mo 2533  df-eu 2562  df-cleq 2723  df-clel 2809  df-rex 3070  df-rmo 3375  df-reu 3376

This theorem is referenced by:  icoshftf1o  13458  addsq2reu  27286  euoreqb  46276  inlinecirc02preu  47636