Theorem intssuni 4931

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 4456	. . . 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 3461	. . . 4 ⊢ 𝑥 ∈ V
4	3	elint2 4915	. . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5		eluni2 4872	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	2, 4, 5	3imtr4g 299	. 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7	6	ssrdv 3945	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907  ∅c0 4288  ∪ cuni 4868  ∩ cint 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-v 3459  df-dif 3910  df-ss 3924  df-nul 4289  df-uni 4869  df-int 4909

This theorem is referenced by:  unissint  4933  intssuni2  4934  intss2  5070  fin23lem31  10315  wunint  10688  tskint  10758  incexc  15881  incexc2  15882  subgint  19208  efgval  19778  lbsextlem3  21253  ssdifidllem  21444  cssmre  21803  uffixfr  24041  uffix2  24042  uffixsn  24043  ssmxidllem  33673  insiga  34444  dfon2lem8  36151  intidl  38540  elrfi  43287  toplatglb  49630