MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relint Structured version   Visualization version   GIF version

Theorem relint 5689
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 5687 . 2 (∃𝑥𝐴 Rel 𝑥 → Rel 𝑥𝐴 𝑥)
2 intiin 4968 . . 3 𝐴 = 𝑥𝐴 𝑥
32releqi 5649 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
41, 3sylibr 237 1 (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3062   cint 4859   ciin 4905  Rel wrel 5556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-in 3873  df-ss 3883  df-int 4860  df-iin 4907  df-rel 5558
This theorem is referenced by:  clrellem  40909
  Copyright terms: Public domain W3C validator