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Theorem relintabex 40076
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 5217 . . . 4 {𝑥𝜑} ∈ V ↔ {𝑥𝜑} = V)
2 nrelv 5649 . . . . 5 ¬ Rel V
3 releq 5627 . . . . 5 ( {𝑥𝜑} = V → (Rel {𝑥𝜑} ↔ Rel V))
42, 3mtbiri 329 . . . 4 ( {𝑥𝜑} = V → ¬ Rel {𝑥𝜑})
51, 4sylbi 219 . . 3 {𝑥𝜑} ∈ V → ¬ Rel {𝑥𝜑})
65con4i 114 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} ∈ V)
7 intexab 5218 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
86, 7sylibr 236 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wex 1780  wcel 2114  {cab 2798  Vcvv 3473   cint 4852  Rel wrel 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-int 4853  df-opab 5105  df-xp 5537  df-rel 5538
This theorem is referenced by:  relintab  40078
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