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Theorem intssuni 4897
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 4439 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥𝑦) → ∃𝑦𝐴 𝑥𝑦)
21ex 415 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥𝑦 → ∃𝑦𝐴 𝑥𝑦))
3 vex 3497 . . . 4 𝑥 ∈ V
43elint2 4882 . . 3 (𝑥 𝐴 ↔ ∀𝑦𝐴 𝑥𝑦)
5 eluni2 4841 . . 3 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
62, 4, 53imtr4g 298 . 2 (𝐴 ≠ ∅ → (𝑥 𝐴𝑥 𝐴))
76ssrdv 3972 1 (𝐴 ≠ ∅ → 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wne 3016  wral 3138  wrex 3139  wss 3935  c0 4290   cuni 4837   cint 4875
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291  df-uni 4838  df-int 4876
This theorem is referenced by:  unissint  4899  intssuni2  4900  fin23lem31  9764  wunint  10136  tskint  10206  incexc  15191  incexc2  15192  subgint  18302  efgval  18842  lbsextlem3  19931  cssmre  20836  uffixfr  22530  uffix2  22531  uffixsn  22532  ssmxidllem  30978  insiga  31396  dfon2lem8  33035  bj-intss  34390  intidl  35306  elrfi  39291
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