Theorem elinsn 4671

Description: If the intersection of two classes is a (proper) singleton, then the singleton element is a member of both classes. (Contributed by AV, 30-Dec-2021.)

Assertion

Ref	Expression
elinsn	⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Proof of Theorem elinsn

Step	Hyp	Ref	Expression
1		snidg 4621	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		eleq2 2850	. . 3 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ 𝐴 ∈ {𝐴}))
3		elin 3915	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	3	biimpi 219	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
5	2, 4	biimtrrdi 257	. 2 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))
6	1, 5	mpan9 516	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898  {csn 4584

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-in 3906  df-sn 4585

This theorem is used by:  frgrncvvdeqlem3  30902  frgrncvvdeqlem6  30905