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Mirrors > Home > MPE Home > Th. List > sscon | Structured version Visualization version GIF version |
Description: Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
sscon | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3970 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | con3d 152 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
3 | 2 | anim2d 610 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))) |
4 | eldif 3954 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) | |
5 | eldif 3954 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4g 295 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐶 ∖ 𝐵) → 𝑥 ∈ (𝐶 ∖ 𝐴))) |
7 | 6 | ssrdv 3982 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ∖ cdif 3941 ⊆ wss 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-dif 3947 df-ss 3961 |
This theorem is referenced by: sscond 4138 complss 4143 sscon34b 4293 sorpsscmpl 7740 sbthlem1 9108 sbthlem2 9109 cantnfp1lem1 9703 cantnfp1lem3 9705 isf34lem7 10404 isf34lem6 10405 setsres 17150 mplsubglem 21961 cctop 22953 clsval2 22998 ntrss 23003 hauscmplem 23354 ptbasin 23525 cfinfil 23841 csdfil 23842 uniioombllem5 25560 kur14lem6 34949 bj-2upln1upl 36631 dvasin 37305 clsk3nimkb 43609 fourierdlem62 45691 caragendifcl 46037 |
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