Theorem uniss 3913

Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
uniss	⊢ (A ⊆ B → ∪A ⊆ ∪B)

Proof of Theorem uniss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . 5 ⊢ (A ⊆ B → (y ∈ A → y ∈ B))
2	1	anim2d 548	. . . 4 ⊢ (A ⊆ B → ((x ∈ y ∧ y ∈ A) → (x ∈ y ∧ y ∈ B)))
3	2	eximdv 1622	. . 3 ⊢ (A ⊆ B → (∃y(x ∈ y ∧ y ∈ A) → ∃y(x ∈ y ∧ y ∈ B)))
4		eluni 3895	. . 3 ⊢ (x ∈ ∪A ↔ ∃y(x ∈ y ∧ y ∈ A))
5		eluni 3895	. . 3 ⊢ (x ∈ ∪B ↔ ∃y(x ∈ y ∧ y ∈ B))
6	3, 4, 5	3imtr4g 261	. 2 ⊢ (A ⊆ B → (x ∈ ∪A → x ∈ ∪B))
7	6	ssrdv 3279	1 ⊢ (A ⊆ B → ∪A ⊆ ∪B)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ⊆ wss 3258  ∪cuni 3892

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is used by:  unissi  3915  unissd  3916  unidif  3924  intssuni2  3952  uniintsn  3964  sspw1  4336