Theorem intssuni 3949

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 3640	. . . 4 ⊢ ((A ≠ ∅ ∧ ∀y ∈ A x ∈ y) → ∃y ∈ A x ∈ y)
2	1	ex 423	. . 3 ⊢ (A ≠ ∅ → (∀y ∈ A x ∈ y → ∃y ∈ A x ∈ y))
3		vex 2863	. . . 4 ⊢ x ∈ V
4	3	elint2 3934	. . 3 ⊢ (x ∈ ∩A ↔ ∀y ∈ A x ∈ y)
5		eluni2 3896	. . 3 ⊢ (x ∈ ∪A ↔ ∃y ∈ A x ∈ y)
6	2, 4, 5	3imtr4g 261	. 2 ⊢ (A ≠ ∅ → (x ∈ ∩A → x ∈ ∪A))
7	6	ssrdv 3279	1 ⊢ (A ≠ ∅ → ∩A ⊆ ∪A)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616   ⊆ wss 3258  ∅c0 3551  ∪cuni 3892  ∩cint 3927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-uni 3893  df-int 3928

This theorem is used by:  unissint  3951  intssuni2  3952