| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 3977 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 2 | 1 | con3d 152 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
| 3 | 2 | anim2d 612 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))) |
| 4 | eldif 3961 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) | |
| 5 | eldif 3961 | . . 3 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) | |
| 6 | 3, 4, 5 | 3imtr4g 296 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐶 ∖ 𝐵) → 𝑥 ∈ (𝐶 ∖ 𝐴))) |
| 7 | 6 | ssrdv 3989 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-ss 3968 |
| This theorem is referenced by: sscond 4146 complss 4151 sscon34b 4304 sorpsscmpl 7754 sbthlem1 9123 sbthlem2 9124 cantnfp1lem1 9718 cantnfp1lem3 9720 isf34lem7 10419 isf34lem6 10420 setsres 17215 mplsubglem 22019 cctop 23013 clsval2 23058 ntrss 23063 hauscmplem 23414 ptbasin 23585 cfinfil 23901 csdfil 23902 uniioombllem5 25622 kur14lem6 35216 bj-2upln1upl 37025 dvasin 37711 readvrec2 42391 clsk3nimkb 44053 fourierdlem62 46183 caragendifcl 46529 |
| Copyright terms: Public domain | W3C validator |