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Theorem relint 5390
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 5388 . 2 (∃𝑥𝐴 Rel 𝑥 → Rel 𝑥𝐴 𝑥)
2 intiin 4718 . . 3 𝐴 = 𝑥𝐴 𝑥
32releqi 5351 . 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 3043   cint 4619   ciin 4665  Rel wrel 5263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-in 3714  df-ss 3721  df-int 4620  df-iin 4667  df-rel 5265
This theorem is referenced by:  clrellem  38423
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