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Mirrors > Home > ILE Home > Th. List > elin2d | 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 |
---|---|
elin2d | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin1d.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | |
2 | elinel2 3294 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 ∩ cin 3101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 |
This theorem is referenced by: elfi2 6917 fiuni 6923 fifo 6925 explecnv 11406 nninfdclemp1 12223 restbasg 12610 txcnp 12713 blin2 12874 bj-charfun 13424 bj-charfundc 13425 |
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