Theorem iunss1 3981

Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss1	⊢ (A ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ B C)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iunss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 3332	. . 3 ⊢ (A ⊆ B → (∃x ∈ A y ∈ C → ∃x ∈ B y ∈ C))
2		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
3		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ B C ↔ ∃x ∈ B y ∈ C)
4	1, 2, 3	3imtr4g 261	. 2 ⊢ (A ⊆ B → (y ∈ ∪x ∈ A C → y ∈ ∪x ∈ B C))
5	4	ssrdv 3279	1 ⊢ (A ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ B C)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710  ∃wrex 2616   ⊆ wss 3258  ∪ciun 3970

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-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 used by:  iuneq1  3983  iunxdif2  4015