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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 3908 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | con3d 155 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
3 | 2 | anim2d 614 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))) |
4 | eldif 3891 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) | |
5 | eldif 3891 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) | |
6 | 3, 4, 5 | 3imtr4g 299 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐶 ∖ 𝐵) → 𝑥 ∈ (𝐶 ∖ 𝐴))) |
7 | 6 | ssrdv 3921 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 |
This theorem is referenced by: sscond 4069 complss 4074 sscon34b 4219 sorpsscmpl 7440 sbthlem1 8611 sbthlem2 8612 cantnfp1lem1 9125 cantnfp1lem3 9127 isf34lem7 9790 isf34lem6 9791 setsres 16517 mplsubglem 20672 cctop 21611 clsval2 21655 ntrss 21660 hauscmplem 22011 ptbasin 22182 cfinfil 22498 csdfil 22499 uniioombllem5 24191 kur14lem6 32571 bj-2upln1upl 34460 dvasin 35141 clsk3nimkb 40743 fourierdlem62 42810 caragendifcl 43153 |
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