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

Theorem relint 5398
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 5396 . 2 (∃𝑥𝐴 Rel 𝑥 → Rel 𝑥𝐴 𝑥)
2 intiin 4726 . . 3 𝐴 = 𝑥𝐴 𝑥
32releqi 5359 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
41, 3sylibr 224 1 (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3051   cint 4627   ciin 4673  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-int 4628  df-iin 4675  df-rel 5273
This theorem is referenced by:  clrellem  38449
  Copyright terms: Public domain W3C validator