Theorem reueq 3703

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 3243	. 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵)
2		moeq 3673	. . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵
3		mormo 3377	. . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)
4	2, 3	ax-mp 5	. . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵
5		reu5 3374	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵))
6	4, 5	mpbiran2 723	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵)
7	1, 6	bitr4i 281	1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∃*wmo 2568  ∃wrex 3092  ∃!wreu 3370  ∃*wrmo 3371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2570  df-eu 2600  df-cleq 2758  df-clel 2841  df-rex 3093  df-rmo 3372  df-reu 3373

This theorem is used by:  icoshftf1o  13519  addsq2reu  27634  euoreqb  47879  isuspgrimlem  48693  inlinecirc02preu  49601