Theorem elinsn 4626

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 4575	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		eleq2 2826	. . 3 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ 𝐴 ∈ {𝐴}))
3		elin 3882	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	3	biimpi 219	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
5	2, 4	syl6bir 257	. 2 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))
6	1, 5	mpan9 510	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ∩ cin 3865  {csn 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-in 3873  df-sn 4542

This theorem is referenced by:  frgrncvvdeqlem3  28384  frgrncvvdeqlem6  28387