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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 4501 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥𝑦) → ∃𝑦𝐴 𝑥𝑦)
21ex 412 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥𝑦 → ∃𝑦𝐴 𝑥𝑦))
3 vex 3482 . . . 4 𝑥 ∈ V
43elint2 4958 . . 3 (𝑥 𝐴 ↔ ∀𝑦𝐴 𝑥𝑦)
5 eluni2 4916 . . 3 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
62, 4, 53imtr4g 296 . 2 (𝐴 ≠ ∅ → (𝑥 𝐴𝑥 𝐴))
76ssrdv 4001 1 (𝐴 ≠ ∅ → 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339   cuni 4912   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-uni 4913  df-int 4952
This theorem is referenced by:  unissint  4977  intssuni2  4978  intss2  5113  fin23lem31  10381  wunint  10753  tskint  10823  incexc  15870  incexc2  15871  subgint  19181  efgval  19750  lbsextlem3  21180  cssmre  21729  uffixfr  23947  uffix2  23948  uffixsn  23949  ssdifidllem  33464  ssmxidllem  33481  insiga  34118  dfon2lem8  35772  intidl  38016  elrfi  42682  toplatglb  48790
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