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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 4128 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | relss 5783 | . 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 3941 ⊆ wss 3944 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-dif 3947 df-ss 3961 df-rel 5685 |
This theorem is referenced by: difopab 5832 difopabOLD 5833 fundif 6603 relsdom 8971 opeldifid 32468 gsumhashmul 32860 fundmpss 35493 relbigcup 35624 vvdifopab 37862 |
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