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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 4090 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | relss 5734 | . 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 3906 ⊆ wss 3909 Rel wrel 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3446 df-dif 3912 df-in 3916 df-ss 3926 df-rel 5638 |
This theorem is referenced by: difopab 5783 difopabOLD 5784 fundif 6546 relsdom 8824 opeldifid 31321 gsumhashmul 31699 fundmpss 34135 relbigcup 34413 vvdifopab 36651 |
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