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Theorem relintabex 43076
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 5335 . . . 4 {𝑥𝜑} ∈ V ↔ {𝑥𝜑} = V)
2 nrelv 5796 . . . . 5 ¬ Rel V
3 releq 5772 . . . . 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 5336 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
86, 7sylibr 233 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wex 1773  wcel 2098  {cab 2702  Vcvv 3463   cint 4944  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-int 4945  df-opab 5206  df-xp 5678  df-rel 5679
This theorem is referenced by:  relintab  43078
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