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Theorem elinsn 4389
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 4351 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2828 . . 3 ((𝐵𝐶) = {𝐴} → (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ {𝐴}))
3 elin 3939 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
43biimpi 206 . . 3 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
52, 4syl6bir 244 . 2 ((𝐵𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴𝐵𝐴𝐶)))
61, 5mpan9 487 1 ((𝐴𝑉 ∧ (𝐵𝐶) = {𝐴}) → (𝐴𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cin 3714  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-sn 4322
This theorem is referenced by:  frgrncvvdeqlem3  27476  frgrncvvdeqlem6  27479
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