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Mirrors > Home > ILE Home > Th. List > unssin | GIF version |
Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Ref | Expression |
---|---|
unssin | ⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oranim 770 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | eldifn 3194 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴) | |
3 | eldifn 3194 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵) | |
4 | 2, 3 | anim12i 336 | . . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
5 | 1, 4 | nsyl 617 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵))) |
6 | elin 3254 | . . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵))) | |
7 | 5, 6 | sylnibr 666 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
8 | elun 3212 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
9 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | eldif 3075 | . . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))) | |
11 | 9, 10 | mpbiran 924 | . . 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 3096 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 697 ∈ wcel 1480 Vcvv 2681 ∖ cdif 3063 ∪ cun 3064 ∩ cin 3065 ⊆ wss 3066 |
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 603 ax-in2 604 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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 |
This theorem is referenced by: (None) |
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