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Mirrors > Home > NFE Home > Th. List > sscon | 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 3267 | . . . . 5 ⊢ (A ⊆ B → (x ∈ A → x ∈ B)) | |
2 | 1 | con3d 125 | . . . 4 ⊢ (A ⊆ B → (¬ x ∈ B → ¬ x ∈ A)) |
3 | 2 | anim2d 548 | . . 3 ⊢ (A ⊆ B → ((x ∈ C ∧ ¬ x ∈ B) → (x ∈ C ∧ ¬ x ∈ A))) |
4 | eldif 3221 | . . 3 ⊢ (x ∈ (C ∖ B) ↔ (x ∈ C ∧ ¬ x ∈ B)) | |
5 | eldif 3221 | . . 3 ⊢ (x ∈ (C ∖ A) ↔ (x ∈ C ∧ ¬ x ∈ A)) | |
6 | 3, 4, 5 | 3imtr4g 261 | . 2 ⊢ (A ⊆ B → (x ∈ (C ∖ B) → x ∈ (C ∖ A))) |
7 | 6 | ssrdv 3278 | 1 ⊢ (A ⊆ B → (C ∖ B) ⊆ (C ∖ A)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 ∈ wcel 1710 ∖ cdif 3206 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 |
This theorem is referenced by: sscond 3403 |
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