Theorem ssundif 3634

Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)

Assertion

Ref	Expression
ssundif	⊢ (A ⊆ (B ∪ C) ↔ (A ∖ B) ⊆ C)

Proof of Theorem ssundif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.6 878	. . . 4 ⊢ (((x ∈ A ∧ ¬ x ∈ B) → x ∈ C) ↔ (x ∈ A → (x ∈ B ∨ x ∈ C)))
2		eldif 3222	. . . . 5 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
3	2	imbi1i 315	. . . 4 ⊢ ((x ∈ (A ∖ B) → x ∈ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) → x ∈ C))
4		elun 3221	. . . . 5 ⊢ (x ∈ (B ∪ C) ↔ (x ∈ B ∨ x ∈ C))
5	4	imbi2i 303	. . . 4 ⊢ ((x ∈ A → x ∈ (B ∪ C)) ↔ (x ∈ A → (x ∈ B ∨ x ∈ C)))
6	1, 3, 5	3bitr4ri 269	. . 3 ⊢ ((x ∈ A → x ∈ (B ∪ C)) ↔ (x ∈ (A ∖ B) → x ∈ C))
7	6	albii 1566	. 2 ⊢ (∀x(x ∈ A → x ∈ (B ∪ C)) ↔ ∀x(x ∈ (A ∖ B) → x ∈ C))
8		dfss2 3263	. 2 ⊢ (A ⊆ (B ∪ C) ↔ ∀x(x ∈ A → x ∈ (B ∪ C)))
9		dfss2 3263	. 2 ⊢ ((A ∖ B) ⊆ C ↔ ∀x(x ∈ (A ∖ B) → x ∈ C))
10	7, 8, 9	3bitr4i 268	1 ⊢ (A ⊆ (B ∪ C) ↔ (A ∖ B) ⊆ C)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∖ cdif 3207   ∪ cun 3208   ⊆ wss 3258

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260

This theorem is referenced by:  difcom  3635  uneqdifeq  3639  ssunsn2  3866  pwadjoin  4120