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Mirrors > Home > ILE Home > Th. List > inssdif | GIF version |
Description: Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.) |
Ref | Expression |
---|---|
inssdif | ⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elndif 3166 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵)) | |
2 | 1 | anim2i 337 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵))) |
3 | elin 3225 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | eldif 3046 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵))) | |
5 | 2, 3, 4 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵))) |
6 | 5 | ssriv 3067 | 1 ⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∈ wcel 1463 Vcvv 2657 ∖ cdif 3034 ∩ cin 3036 ⊆ wss 3037 |
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-in1 586 ax-in2 587 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-dif 3039 df-in 3043 df-ss 3050 |
This theorem is referenced by: difdif2ss 3299 |
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