Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3259 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 413 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∩ cin 3070 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 |
This theorem is referenced by: elfir 6861 infpwfidom 7054 strslfv2d 12001 baspartn 12217 bastg 12230 isopn3 12294 restbasg 12337 lmss 12415 metrest 12675 tgioo 12715 dvmulxxbr 12835 pilem3 12864 |
Copyright terms: Public domain | W3C validator |