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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 3961 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | con3d 155 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
3 | 2 | anim2d 613 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))) |
4 | eldif 3946 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) | |
5 | eldif 3946 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4g 298 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐶 ∖ 𝐵) → 𝑥 ∈ (𝐶 ∖ 𝐴))) |
7 | 6 | ssrdv 3973 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ∖ cdif 3933 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-in 3943 df-ss 3952 |
This theorem is referenced by: sscond 4118 complss 4123 sorpsscmpl 7454 sbthlem1 8621 sbthlem2 8622 cantnfp1lem1 9135 cantnfp1lem3 9137 isf34lem7 9795 isf34lem6 9796 setsres 16519 mplsubglem 20208 cctop 21608 clsval2 21652 ntrss 21657 hauscmplem 22008 ptbasin 22179 cfinfil 22495 csdfil 22496 uniioombllem5 24182 kur14lem6 32453 bj-2upln1upl 34331 dvasin 34972 sscon34b 40362 clsk3nimkb 40383 fourierdlem62 42446 caragendifcl 42789 |
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