MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intssuni Structured version   Visualization version   GIF version

Theorem intssuni 4889
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 4436 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥𝑦) → ∃𝑦𝐴 𝑥𝑦)
21ex 413 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥𝑦 → ∃𝑦𝐴 𝑥𝑦))
3 vex 3495 . . . 4 𝑥 ∈ V
43elint2 4874 . . 3 (𝑥 𝐴 ↔ ∀𝑦𝐴 𝑥𝑦)
5 eluni2 4834 . . 3 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
62, 4, 53imtr4g 297 . 2 (𝐴 ≠ ∅ → (𝑥 𝐴𝑥 𝐴))
76ssrdv 3970 1 (𝐴 ≠ ∅ → 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wne 3013  wral 3135  wrex 3136  wss 3933  c0 4288   cuni 4830   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-uni 4831  df-int 4868
This theorem is referenced by:  unissint  4891  intssuni2  4892  fin23lem31  9753  wunint  10125  tskint  10195  incexc  15180  incexc2  15181  subgint  18241  efgval  18772  lbsextlem3  19861  cssmre  20765  uffixfr  22459  uffix2  22460  uffixsn  22461  insiga  31295  dfon2lem8  32932  bj-intss  34285  intidl  35188  elrfi  39169
  Copyright terms: Public domain W3C validator