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Theorem elinsn 4713
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 ((𝐴𝑉 ∧ (𝐵𝐶) = {𝐴}) → (𝐴𝐵𝐴𝐶))

Proof of Theorem elinsn
StepHypRef Expression
1 snidg 4661 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2820 . . 3 ((𝐵𝐶) = {𝐴} → (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ {𝐴}))
3 elin 3963 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
43biimpi 215 . . 3 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
52, 4syl6bir 253 . 2 ((𝐵𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴𝐵𝐴𝐶)))
61, 5mpan9 505 1 ((𝐴𝑉 ∧ (𝐵𝐶) = {𝐴}) → (𝐴𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  cin 3946  {csn 4627
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-sn 4628
This theorem is referenced by:  frgrncvvdeqlem3  29821  frgrncvvdeqlem6  29824
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