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Theorem relintabex 43594
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 5810 . . . . 5 ¬ Rel V
3 releq 5786 . . . . 5 ( {𝑥𝜑} = V → (Rel {𝑥𝜑} ↔ Rel V))
42, 3mtbiri 327 . . . 4 ( {𝑥𝜑} = V → ¬ Rel {𝑥𝜑})
51, 4sylbi 217 . . 3 {𝑥𝜑} ∈ V → ¬ Rel {𝑥𝜑})
65con4i 114 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} ∈ V)
7 intexab 5346 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
86, 7sylibr 234 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wex 1779  wcel 2108  {cab 2714  Vcvv 3480   cint 4946  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-int 4947  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  relintab  43596
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