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Mirrors > Home > NFE Home > Th. List > sscon34 | GIF version |
Description: Contraposition law for subset. (Contributed by SF, 11-Mar-2015.) |
Ref | Expression |
---|---|
sscon34 | ⊢ (A ⊆ B ↔ ∼ B ⊆ ∼ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con34b 283 | . . . 4 ⊢ ((x ∈ A → x ∈ B) ↔ (¬ x ∈ B → ¬ x ∈ A)) | |
2 | vex 2862 | . . . . . 6 ⊢ x ∈ V | |
3 | 2 | elcompl 3225 | . . . . 5 ⊢ (x ∈ ∼ B ↔ ¬ x ∈ B) |
4 | 2 | elcompl 3225 | . . . . 5 ⊢ (x ∈ ∼ A ↔ ¬ x ∈ A) |
5 | 3, 4 | imbi12i 316 | . . . 4 ⊢ ((x ∈ ∼ B → x ∈ ∼ A) ↔ (¬ x ∈ B → ¬ x ∈ A)) |
6 | 1, 5 | bitr4i 243 | . . 3 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ ∼ B → x ∈ ∼ A)) |
7 | 6 | albii 1566 | . 2 ⊢ (∀x(x ∈ A → x ∈ B) ↔ ∀x(x ∈ ∼ B → x ∈ ∼ A)) |
8 | dfss2 3262 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
9 | dfss2 3262 | . 2 ⊢ ( ∼ B ⊆ ∼ A ↔ ∀x(x ∈ ∼ B → x ∈ ∼ A)) | |
10 | 7, 8, 9 | 3bitr4i 268 | 1 ⊢ (A ⊆ B ↔ ∼ B ⊆ ∼ A) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∀wal 1540 ∈ wcel 1710 ∼ ccompl 3205 ⊆ 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-ss 3259 |
This theorem is referenced by: sbthlem1 6203 |
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