Theorem unss 3438

Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)

Assertion

Ref	Expression
unss	⊢ ((A ⊆ C ∧ B ⊆ C) ↔ (A ∪ B) ⊆ C)

Proof of Theorem unss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3263	. 2 ⊢ ((A ∪ B) ⊆ C ↔ ∀x(x ∈ (A ∪ B) → x ∈ C))
2		19.26 1593	. . 3 ⊢ (∀x((x ∈ A → x ∈ C) ∧ (x ∈ B → x ∈ C)) ↔ (∀x(x ∈ A → x ∈ C) ∧ ∀x(x ∈ B → x ∈ C)))
3		elun 3221	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
4	3	imbi1i 315	. . . . 5 ⊢ ((x ∈ (A ∪ B) → x ∈ C) ↔ ((x ∈ A ∨ x ∈ B) → x ∈ C))
5		jaob 758	. . . . 5 ⊢ (((x ∈ A ∨ x ∈ B) → x ∈ C) ↔ ((x ∈ A → x ∈ C) ∧ (x ∈ B → x ∈ C)))
6	4, 5	bitri 240	. . . 4 ⊢ ((x ∈ (A ∪ B) → x ∈ C) ↔ ((x ∈ A → x ∈ C) ∧ (x ∈ B → x ∈ C)))
7	6	albii 1566	. . 3 ⊢ (∀x(x ∈ (A ∪ B) → x ∈ C) ↔ ∀x((x ∈ A → x ∈ C) ∧ (x ∈ B → x ∈ C)))
8		dfss2 3263	. . . 4 ⊢ (A ⊆ C ↔ ∀x(x ∈ A → x ∈ C))
9		dfss2 3263	. . . 4 ⊢ (B ⊆ C ↔ ∀x(x ∈ B → x ∈ C))
10	8, 9	anbi12i 678	. . 3 ⊢ ((A ⊆ C ∧ B ⊆ C) ↔ (∀x(x ∈ A → x ∈ C) ∧ ∀x(x ∈ B → x ∈ C)))
11	2, 7, 10	3bitr4i 268	. 2 ⊢ (∀x(x ∈ (A ∪ B) → x ∈ C) ↔ (A ⊆ C ∧ B ⊆ C))
12	1, 11	bitr2i 241	1 ⊢ ((A ⊆ C ∧ B ⊆ C) ↔ (A ∪ B) ⊆ C)

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∪ 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-ss 3260

This theorem is referenced by:  unssi  3439  unssd  3440  unssad  3441  unssbd  3442  nsspssun  3489  uneqin  3507  uneqdifeq  3639  prss  3862  prssg  3863  ssunsn2  3866  tpss  3872  evenoddnnnul  4515  nnadjoinpw  4522  tfinnn  4535  vfinspeqtncv  4554  sbthlem1  6204  spacssnc  6285