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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 4146 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | relss 5794 | . 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 3960 ⊆ wss 3963 Rel wrel 5694 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-ss 3980 df-rel 5696 |
This theorem is referenced by: difopab 5843 difopabOLD 5844 fundif 6617 relsdom 8991 opeldifid 32619 gsumhashmul 33047 fundmpss 35748 relbigcup 35879 vvdifopab 38242 |
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