Theorem elinsn 4676

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 4626	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		eleq2 2818	. . 3 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ 𝐴 ∈ {𝐴}))
3		elin 3932	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	3	biimpi 216	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
5	2, 4	biimtrrdi 254	. 2 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))
6	1, 5	mpan9 506	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3915  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3923  df-sn 4592

This theorem is referenced by:  frgrncvvdeqlem3  30236  frgrncvvdeqlem6  30239