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Theorem intssuni 4975
Description: The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssuni (𝐴 ≠ ∅ → 𝐴 𝐴)

Proof of Theorem intssuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.2z 4495 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥𝑦) → ∃𝑦𝐴 𝑥𝑦)
21ex 414 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥𝑦 → ∃𝑦𝐴 𝑥𝑦))
3 vex 3479 . . . 4 𝑥 ∈ V
43elint2 4958 . . 3 (𝑥 𝐴 ↔ ∀𝑦𝐴 𝑥𝑦)
5 eluni2 4913 . . 3 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
62, 4, 53imtr4g 296 . 2 (𝐴 ≠ ∅ → (𝑥 𝐴𝑥 𝐴))
76ssrdv 3989 1 (𝐴 ≠ ∅ → 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 2941  wral 3062  wrex 3071  wss 3949  c0 4323   cuni 4909   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-uni 4910  df-int 4952
This theorem is referenced by:  unissint  4977  intssuni2  4978  intss2  5112  fin23lem31  10338  wunint  10710  tskint  10780  incexc  15783  incexc2  15784  subgint  19030  efgval  19585  lbsextlem3  20773  cssmre  21246  uffixfr  23427  uffix2  23428  uffixsn  23429  ssmxidllem  32589  insiga  33135  dfon2lem8  34762  intidl  36897  elrfi  41432  toplatglb  47626
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