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Mirrors > Home > ILE Home > Th. List > ineqri | GIF version |
Description: Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ineqri.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) |
Ref | Expression |
---|---|
ineqri | ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3225 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | ineqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 183 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2112 | 1 ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1314 ∈ wcel 1463 ∩ cin 3036 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-in 3043 |
This theorem is referenced by: inidm 3251 inass 3252 indi 3289 inab 3310 in0 3363 pwin 4164 dmres 4798 |
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