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Theorem intssuni 4531
 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 4093 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥𝑦) → ∃𝑦𝐴 𝑥𝑦)
21ex 449 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥𝑦 → ∃𝑦𝐴 𝑥𝑦))
3 vex 3234 . . . 4 𝑥 ∈ V
43elint2 4514 . . 3 (𝑥 𝐴 ↔ ∀𝑦𝐴 𝑥𝑦)
5 eluni2 4472 . . 3 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
62, 4, 53imtr4g 285 . 2 (𝐴 ≠ ∅ → (𝑥 𝐴𝑥 𝐴))
76ssrdv 3642 1 (𝐴 ≠ ∅ → 𝐴 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  ∅c0 3948  ∪ cuni 4468  ∩ cint 4507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-uni 4469  df-int 4508 This theorem is referenced by:  unissint  4533  intssuni2  4534  fin23lem31  9203  wunint  9575  tskint  9645  incexc  14613  incexc2  14614  subgint  17665  efgval  18176  lbsextlem3  19208  cssmre  20085  uffixfr  21774  uffix2  21775  uffixsn  21776  insiga  30328  dfon2lem8  31819  bj-intss  33178  intidl  33958  elrfi  37574
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