Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elin2 | GIF version |
Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
elin2.x | ⊢ 𝑋 = (𝐵 ∩ 𝐶) |
Ref | Expression |
---|---|
elin2 | ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin2.x | . . 3 ⊢ 𝑋 = (𝐵 ∩ 𝐶) | |
2 | 1 | eleq2i 2237 | . 2 ⊢ (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝐵 ∩ 𝐶)) |
3 | elin 3310 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ 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: elin3 3318 fnres 5314 funfvima 5727 lmres 13042 |
Copyright terms: Public domain | W3C validator |