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Mirrors > Home > NFE Home > Th. List > ssdifss | GIF version |
Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.) |
Ref | Expression |
---|---|
ssdifss | ⊢ (A ⊆ B → (A ∖ C) ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3393 | . 2 ⊢ (A ∖ C) ⊆ A | |
2 | sstr 3280 | . 2 ⊢ (((A ∖ C) ⊆ A ∧ A ⊆ B) → (A ∖ C) ⊆ B) | |
3 | 1, 2 | mpan 651 | 1 ⊢ (A ⊆ B → (A ∖ C) ⊆ B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ 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: ssdifssd 3404 |
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