Theorem unssi 3439

Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Hypotheses

Ref	Expression
unssi.1	⊢ A ⊆ C
unssi.2	⊢ B ⊆ C

Assertion

Ref	Expression
unssi	⊢ (A ∪ B) ⊆ C

Proof of Theorem unssi

Step	Hyp	Ref	Expression
1		unssi.1	. . 3 ⊢ A ⊆ C
2		unssi.2	. . 3 ⊢ B ⊆ C
3	1, 2	pm3.2i 441	. 2 ⊢ (A ⊆ C ∧ B ⊆ C)
4		unss 3438	. 2 ⊢ ((A ⊆ C ∧ B ⊆ C) ↔ (A ∪ B) ⊆ C)
5	3, 4	mpbi 199	1 ⊢ (A ∪ B) ⊆ C

Syntax hints:   ∧ wa 358   ∪ 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: (None)