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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 807 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
2 | eldifn 3067 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴) | |
3 | eldifn 3067 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵) | |
4 | 2, 3 | anim12i 321 | . . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
5 | 1, 4 | nsyl 558 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵))) |
6 | elin 3126 | . . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵))) | |
7 | 5, 6 | sylnibr 602 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
8 | elun 3084 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
9 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | eldif 2927 | . . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))) | |
11 | 9, 10 | mpbiran 847 | . . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
12 | 7, 8, 11 | 3imtr4i 190 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))) |
13 | 12 | ssriv 2949 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ∨ wo 629 ∈ wcel 1393 Vcvv 2557 ∖ cdif 2914 ∪ cun 2915 ∩ cin 2916 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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