Theorem intssuni 4934

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.

Step	Hyp	Ref	Expression
1		r19.2z 4459	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2	1	ex 417	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
3		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
4	3	elint2 4918	. . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5		eluni2 4875	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	2, 4, 5	3imtr4g 299	. 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7	6	ssrdv 3942	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4871  ∩ cint 4911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-uni 4872  df-int 4912

This theorem is used by:  unissint  4936  intssuni2  4937  intss2  5073  fin23lem31  10333  wunint  10706  tskint  10776  incexc  15898  incexc2  15899  subgint  19223  efgval  19793  lbsextlem3  21295  ssdifidllem  21495  cssmre  21854  uffixfr  24091  uffix2  24092  uffixsn  24093  ssmxidllem  33765  insiga  34536  dfon2lem8  36288  intidl  38708  elrfi  43453  toplatglb  49807