Theorem ssiun 4009

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun	⊢ (∃x ∈ A C ⊆ B → C ⊆ ∪x ∈ A B)

Distinct variable group:   x,C

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem ssiun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . 5 ⊢ (C ⊆ B → (y ∈ C → y ∈ B))
2	1	reximi 2722	. . . 4 ⊢ (∃x ∈ A C ⊆ B → ∃x ∈ A (y ∈ C → y ∈ B))
3		r19.37av 2762	. . . 4 ⊢ (∃x ∈ A (y ∈ C → y ∈ B) → (y ∈ C → ∃x ∈ A y ∈ B))
4	2, 3	syl 15	. . 3 ⊢ (∃x ∈ A C ⊆ B → (y ∈ C → ∃x ∈ A y ∈ B))
5		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
6	4, 5	syl6ibr 218	. 2 ⊢ (∃x ∈ A C ⊆ B → (y ∈ C → y ∈ ∪x ∈ A B))
7	6	ssrdv 3279	1 ⊢ (∃x ∈ A C ⊆ B → C ⊆ ∪x ∈ A B)

Syntax hints:   → wi 4   ∈ wcel 1710  ∃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:  iunss2  4012  iunpwss  4056