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Theorem relintabex 43248
Description: If the intersection of a class is a relation, then the class is nonempty. (Contributed by RP, 12-Aug-2020.)
Assertion
Ref Expression
relintabex (Rel {𝑥𝜑} → ∃𝑥𝜑)

Proof of Theorem relintabex
StepHypRef Expression
1 intnex 5345 . . . 4 {𝑥𝜑} ∈ V ↔ {𝑥𝜑} = V)
2 nrelv 5806 . . . . 5 ¬ Rel V
3 releq 5782 . . . . 5 ( {𝑥𝜑} = V → (Rel {𝑥𝜑} ↔ Rel V))
42, 3mtbiri 326 . . . 4 ( {𝑥𝜑} = V → ¬ Rel {𝑥𝜑})
51, 4sylbi 216 . . 3 {𝑥𝜑} ∈ V → ¬ Rel {𝑥𝜑})
65con4i 114 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} ∈ V)
7 intexab 5346 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
86, 7sylibr 233 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1534  wex 1774  wcel 2099  {cab 2703  Vcvv 3462   cint 4954  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-int 4955  df-opab 5216  df-xp 5688  df-rel 5689
This theorem is referenced by:  relintab  43250
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