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Theorem elinsn 4606
 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 4559 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2878 . . 3 ((𝐵𝐶) = {𝐴} → (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ {𝐴}))
3 elin 3897 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
43biimpi 219 . . 3 (𝐴 ∈ (𝐵𝐶) → (𝐴𝐵𝐴𝐶))
52, 4syl6bir 257 . 2 ((𝐵𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴𝐵𝐴𝐶)))
61, 5mpan9 510 1 ((𝐴𝑉 ∧ (𝐵𝐶) = {𝐴}) → (𝐴𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∩ cin 3880  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-sn 4526 This theorem is referenced by:  frgrncvvdeqlem3  28096  frgrncvvdeqlem6  28099
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