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Mirrors > Home > NFE Home > Th. List > difss2d | GIF version |
Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3396. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difss2d.1 | ⊢ (φ → A ⊆ (B ∖ C)) |
Ref | Expression |
---|---|
difss2d | ⊢ (φ → A ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss2d.1 | . 2 ⊢ (φ → A ⊆ (B ∖ C)) | |
2 | difss2 3396 | . 2 ⊢ (A ⊆ (B ∖ C) → A ⊆ B) | |
3 | 1, 2 | syl 15 | 1 ⊢ (φ → A ⊆ B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3207 ⊆ wss 3258 |
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 2479 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-dif 3216 df-ss 3260 |
This theorem is referenced by: (None) |
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