Theorem ss2iun 2581

Description: Subclass theorem for indexed union.

Assertion

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

Proof of Theorem ss2iun

Step	Hyp	Ref	Expression
1		hbra1 1690	. . . 4 ⊢ (∀x ∈ A B ⊆ C → ∀x∀x ∈ A B ⊆ C)
2		ra4 1697	. . . . 5 ⊢ (∀x ∈ A B ⊆ C → (x ∈ A → B ⊆ C))
3		ssel 2066	. . . . 5 ⊢ (B ⊆ C → (y ∈ B → y ∈ C))
4	2, 3	syl6 22	. . . 4 ⊢ (∀x ∈ A B ⊆ C → (x ∈ A → (y ∈ B → y ∈ C)))
5	1, 4	r19.22d 1738	. . 3 ⊢ (∀x ∈ A B ⊆ C → (∃x ∈ A y ∈ B → ∃x ∈ A y ∈ C))
6		eliun 2574	. . 3 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
7		eliun 2574	. . 3 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
8	5, 6, 7	3imtr4g 555	. 2 ⊢ (∀x ∈ A B ⊆ C → (y ∈ ∪x ∈ A B → y ∈ ∪x ∈ A C))
9	8	ssrdv 2073	1 ⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C)

Syntax hints:   → wi 3   ∈ wcel 960  ∀wral 1648  ∃wrex 1649   ⊆ wss 2050  ∪ciun 2570

This theorem is referenced by:  iuneq2 2582  oawordri 4190  omwordri 4209  oewordri 4225  oeworde 4226  r1val1 4668

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-iun 2572

Copyright terms: Public domain