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Mirrors > Home > ILE Home > Th. List > elind | GIF version |
Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
elind.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
elind.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
elind | ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elind.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
2 | elind.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | elin 3310 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 415 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ∩ cin 3120 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 |
This theorem is referenced by: elfir 6950 infpwfidom 7175 nninfdclemcl 12403 nninfdclemp1 12405 strslfv2d 12458 insubm 12703 baspartn 12842 bastg 12855 isopn3 12919 restbasg 12962 lmss 13040 metrest 13300 tgioo 13340 dvmulxxbr 13460 pilem3 13498 2sqlem7 13751 |
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