Theorem elinsn 4710

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 4658	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		eleq2 2823	. . 3 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ 𝐴 ∈ {𝐴}))
3		elin 3962	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	3	biimpi 215	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
5	2, 4	syl6bir 254	. 2 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))
6	1, 5	mpan9 508	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∩ cin 3945  {csn 4624

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3953  df-sn 4625

This theorem is referenced by:  frgrncvvdeqlem3  29521  frgrncvvdeqlem6  29524