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Mirrors > Home > MPE Home > Th. List > ssdifd | Structured version Visualization version GIF version |
Description: If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 3745. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssdifd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ssdifd | ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdifd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | ssdif 3745 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3571 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 |
This theorem is referenced by: ssdif2d 3749 domunsncan 8060 fin1a2lem13 9234 seqcoll2 13249 rpnnen2lem11 14953 coprmprod 15375 mrieqv2d 16299 mrissmrid 16301 mreexexlem4d 16307 acsfiindd 17177 lsppratlem3 19149 lsppratlem4 19150 f1lindf 20161 lpss3 20948 lpcls 21168 fin1aufil 21736 rrxmval 23188 rrxmetlem 23190 uniioombllem3 23353 i1fmul 23463 itg1addlem4 23466 itg1climres 23481 limciun 23658 ig1peu 23931 ig1pdvds 23936 fusgreghash2wspv 27199 indsumin 30084 sitgclg 30404 mthmpps 31479 poimirlem11 33420 poimirlem12 33421 poimirlem15 33424 dochfln0 36766 lcfl6 36789 lcfrlem16 36847 hdmaprnlem4N 37145 caragendifcl 40728 |
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