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Mirrors > Home > MPE Home > Th. List > reldif | Structured version Visualization version GIF version |
Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
reldif | ⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4107 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | relss 5650 | . 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3932 ⊆ wss 3935 Rel wrel 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-rel 5556 |
This theorem is referenced by: difopab 5696 fundif 6397 relsdom 8510 opeldifid 30343 fundmpss 33004 relbigcup 33353 vvdifopab 35515 |
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