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Theorem relint 5780
Description: The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
Assertion
Ref Expression
relint (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem relint
StepHypRef Expression
1 reliin 5778 . 2 (∃𝑥𝐴 Rel 𝑥 → Rel 𝑥𝐴 𝑥)
2 intiin 5024 . . 3 𝐴 = 𝑥𝐴 𝑥
32releqi 5738 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
41, 3sylibr 233 1 (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3074   cint 4912   ciin 4960  Rel wrel 5643
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-v 3450  df-in 3922  df-ss 3932  df-int 4913  df-iin 4962  df-rel 5645
This theorem is referenced by:  clrellem  41968
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