Theorem intssuni 4970

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 4495	. . . 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 3484	. . . 4 ⊢ 𝑥 ∈ V
4	3	elint2 4953	. . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5		eluni2 4911	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	2, 4, 5	3imtr4g 296	. 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴))
7	6	ssrdv 3989	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907  ∩ cint 4946

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-uni 4908  df-int 4947

This theorem is referenced by:  unissint  4972  intssuni2  4973  intss2  5108  fin23lem31  10383  wunint  10755  tskint  10825  incexc  15873  incexc2  15874  subgint  19168  efgval  19735  lbsextlem3  21162  cssmre  21711  uffixfr  23931  uffix2  23932  uffixsn  23933  ssdifidllem  33484  ssmxidllem  33501  insiga  34138  dfon2lem8  35791  intidl  38036  elrfi  42705  toplatglb  48890