Theorem unss 3437

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 3262	. 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 3220	. . . . . 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 3262	. . . 4 ⊢ (A ⊆ C ↔ ∀x(x ∈ A → x ∈ C))
9		dfss2 3262	. . . 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 3207   ⊆ wss 3257

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259

This theorem is referenced by:  unssi  3438  unssd  3439  unssad  3440  unssbd  3441  nsspssun  3488  uneqin  3506  uneqdifeq  3638  prss  3861  prssg  3862  ssunsn2  3865  tpss  3871  evenoddnnnul  4514  nnadjoinpw  4521  tfinnn  4534  vfinspeqtncv  4553  sbthlem1  6203  spacssnc  6284