Theorem reueq 3672

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 3194	. 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵)
2		moeq 3642	. . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵
3		mormo 3360	. . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)
4	2, 3	ax-mp 5	. . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵
5		reu5 3361	. . 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 277	1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∃*wmo 2538  ∃wrex 3065  ∃!wreu 3066  ∃*wrmo 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-eu 2569  df-cleq 2730  df-clel 2816  df-rex 3070  df-rmo 3071  df-reu 3072

This theorem is referenced by:  icoshftf1o  13206  addsq2reu  26588  euoreqb  44601  inlinecirc02preu  46134