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Mirrors > Home > ILE Home > Th. List > inssun | GIF version |
Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Ref | Expression |
---|---|
inssun | ⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.1 744 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵)) | |
2 | eldifn 3240 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴) | |
3 | eldifn 3240 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵) | |
4 | 2, 3 | orim12i 749 | . . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵)) |
5 | 1, 4 | nsyl 618 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵))) |
6 | elun 3258 | . . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵))) | |
7 | 5, 6 | sylnibr 667 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
8 | elin 3300 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
9 | vex 2724 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | eldif 3120 | . . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))) | |
11 | 9, 10 | mpbiran 929 | . . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
12 | 7, 8, 11 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))) |
13 | 12 | ssriv 3141 | 1 ⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 698 ∈ wcel 2135 Vcvv 2721 ∖ cdif 3108 ∪ cun 3109 ∩ cin 3110 ⊆ wss 3111 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 |
This theorem is referenced by: (None) |
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