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Theorem relint 5153
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 5151 . 2 (∃𝑥𝐴 Rel 𝑥 → Rel 𝑥𝐴 𝑥)
2 intiin 4504 . . 3 𝐴 = 𝑥𝐴 𝑥
32releqi 5114 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
41, 3sylibr 222 1 (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 2896   cint 4404   ciin 4450  Rel wrel 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-int 4405  df-iin 4452  df-rel 5034
This theorem is referenced by:  clrellem  36731
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