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Mirrors > Home > MPE Home > Th. List > elinsn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
elinsn | ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 4593 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | eleq2 2901 | . . 3 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ 𝐴 ∈ {𝐴})) | |
3 | elin 4169 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
4 | 3 | biimpi 218 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
5 | 2, 4 | syl6bir 256 | . 2 ⊢ ((𝐵 ∩ 𝐶) = {𝐴} → (𝐴 ∈ {𝐴} → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) |
6 | 1, 5 | mpan9 509 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3935 {csn 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-in 3943 df-sn 4562 |
This theorem is referenced by: frgrncvvdeqlem3 28074 frgrncvvdeqlem6 28077 |
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