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Theorem intssuni 4968
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 4490 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥𝑦) → ∃𝑦𝐴 𝑥𝑦)
21ex 412 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥𝑦 → ∃𝑦𝐴 𝑥𝑦))
3 vex 3473 . . . 4 𝑥 ∈ V
43elint2 4951 . . 3 (𝑥 𝐴 ↔ ∀𝑦𝐴 𝑥𝑦)
5 eluni2 4907 . . 3 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
62, 4, 53imtr4g 296 . 2 (𝐴 ≠ ∅ → (𝑥 𝐴𝑥 𝐴))
76ssrdv 3984 1 (𝐴 ≠ ∅ → 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wne 2935  wral 3056  wrex 3065  wss 3944  c0 4318   cuni 4903   cint 4944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-uni 4904  df-int 4945
This theorem is referenced by:  unissint  4970  intssuni2  4971  intss2  5105  fin23lem31  10358  wunint  10730  tskint  10800  incexc  15807  incexc2  15808  subgint  19096  efgval  19663  lbsextlem3  21037  cssmre  21612  uffixfr  23814  uffix2  23815  uffixsn  23816  ssmxidllem  33122  insiga  33692  dfon2lem8  35322  intidl  37437  elrfi  42036  toplatglb  47935
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