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Mirrors > Home > MPE Home > Th. List > elin1d | Structured version Visualization version GIF version |
Description: Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.) |
Ref | Expression |
---|---|
elin1d.1 | ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
Ref | Expression |
---|---|
elin1d | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin1d.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | |
2 | elinel1 3832 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 ∩ cin 3606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-v 3233 df-in 3614 |
This theorem is referenced by: nmoleub2lem3 22961 nmoleub3 22965 tayl0 24161 esum2d 30283 ispisys2 30344 sigapisys 30346 sigapildsyslem 30352 sigapildsys 30353 eulerpartlemgvv 30566 tgoldbachgt 30869 |
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