Theorem ssuni 3914

Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
ssuni	⊢ ((A ⊆ B ∧ B ∈ C) → A ⊆ ∪C)

Proof of Theorem ssuni

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

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . . . 7 ⊢ (x = B → (y ∈ x ↔ y ∈ B))
2	1	imbi1d 308	. . . . . 6 ⊢ (x = B → ((y ∈ x → y ∈ ∪C) ↔ (y ∈ B → y ∈ ∪C)))
3		elunii 3897	. . . . . . 7 ⊢ ((y ∈ x ∧ x ∈ C) → y ∈ ∪C)
4	3	expcom 424	. . . . . 6 ⊢ (x ∈ C → (y ∈ x → y ∈ ∪C))
5	2, 4	vtoclga 2921	. . . . 5 ⊢ (B ∈ C → (y ∈ B → y ∈ ∪C))
6	5	imim2d 48	. . . 4 ⊢ (B ∈ C → ((y ∈ A → y ∈ B) → (y ∈ A → y ∈ ∪C)))
7	6	alimdv 1621	. . 3 ⊢ (B ∈ C → (∀y(y ∈ A → y ∈ B) → ∀y(y ∈ A → y ∈ ∪C)))
8		dfss2 3263	. . 3 ⊢ (A ⊆ B ↔ ∀y(y ∈ A → y ∈ B))
9		dfss2 3263	. . 3 ⊢ (A ⊆ ∪C ↔ ∀y(y ∈ A → y ∈ ∪C))
10	7, 8, 9	3imtr4g 261	. 2 ⊢ (B ∈ C → (A ⊆ B → A ⊆ ∪C))
11	10	impcom 419	1 ⊢ ((A ⊆ B ∧ B ∈ C) → A ⊆ ∪C)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ 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:  elssuni  3920  uniss2  3923