Theorem intssuni 4936

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 4457	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2	1	ex 413	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
3		vex 3450	. . . 4 ⊢ 𝑥 ∈ V
4	3	elint2 4919	. . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5		eluni2 4874	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	2, 4, 5	3imtr4g 295	. 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7	6	ssrdv 3953	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ⊆ wss 3913  ∅c0 4287  ∪ cuni 4870  ∩ cint 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3448  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4288  df-uni 4871  df-int 4913

This theorem is referenced by:  unissint  4938  intssuni2  4939  intss2  5073  fin23lem31  10288  wunint  10660  tskint  10730  incexc  15733  incexc2  15734  subgint  18966  efgval  19513  lbsextlem3  20680  cssmre  21134  uffixfr  23311  uffix2  23312  uffixsn  23313  ssmxidllem  32314  insiga  32825  dfon2lem8  34451  intidl  36561  elrfi  41075  toplatglb  47146