Theorem ss2iun 3985

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ss2iun	⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C)

Proof of Theorem ss2iun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . 5 ⊢ (B ⊆ C → (y ∈ B → y ∈ C))
2	1	ralimi 2690	. . . 4 ⊢ (∀x ∈ A B ⊆ C → ∀x ∈ A (y ∈ B → y ∈ C))
3		rexim 2719	. . . 4 ⊢ (∀x ∈ A (y ∈ B → y ∈ C) → (∃x ∈ A y ∈ B → ∃x ∈ A y ∈ C))
4	2, 3	syl 15	. . 3 ⊢ (∀x ∈ A B ⊆ C → (∃x ∈ A y ∈ B → ∃x ∈ A y ∈ C))
5		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
6		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
7	4, 5, 6	3imtr4g 261	. 2 ⊢ (∀x ∈ A B ⊆ C → (y ∈ ∪x ∈ A B → y ∈ ∪x ∈ A C))
8	7	ssrdv 3279	1 ⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C)

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ⊆ wss 3258  ∪ciun 3970

This theorem was proved from 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 theorem 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-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  iuneq2  3986