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Theorem reldif 5661
 Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
reldif (Rel 𝐴 → Rel (𝐴𝐵))

Proof of Theorem reldif
StepHypRef Expression
1 difss 4039 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5629 . 2 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
31, 2ax-mp 5 1 (Rel 𝐴 → Rel (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∖ cdif 3857   ⊆ wss 3860  Rel wrel 5532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3863  df-in 3867  df-ss 3877  df-rel 5534 This theorem is referenced by:  difopab  5676  fundif  6388  relsdom  8539  opeldifid  30465  gsumhashmul  30846  fundmpss  33260  relbigcup  33774  vvdifopab  35987
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