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Theorem reldif 5376
 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 3888 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5345 . 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 3720   ⊆ wss 3723  Rel wrel 5255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-rel 5257 This theorem is referenced by:  difopab  5391  fundif  6077  relsdom  8120  opeldifid  29750  fundmpss  32002  relbigcup  32341  vvdifopab  34367
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