Theorem intssuni 4937

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 4461	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2	1	ex 412	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
3		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
4	3	elint2 4920	. . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5		eluni2 4878	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	2, 4, 5	3imtr4g 296	. 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7	6	ssrdv 3955	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  ∅c0 4299  ∪ cuni 4874  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-v 3452  df-dif 3920  df-ss 3934  df-nul 4300  df-uni 4875  df-int 4914

This theorem is referenced by:  unissint  4939  intssuni2  4940  intss2  5075  fin23lem31  10303  wunint  10675  tskint  10745  incexc  15810  incexc2  15811  subgint  19089  efgval  19654  lbsextlem3  21077  cssmre  21609  uffixfr  23817  uffix2  23818  uffixsn  23819  ssdifidllem  33434  ssmxidllem  33451  insiga  34134  dfon2lem8  35785  intidl  38030  elrfi  42689  toplatglb  48993