Theorem unss2 3434

Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
unss2	⊢ (A ⊆ B → (C ∪ A) ⊆ (C ∪ B))

Proof of Theorem unss2

Step	Hyp	Ref	Expression
1		unss1 3432	. 2 ⊢ (A ⊆ B → (A ∪ C) ⊆ (B ∪ C))
2		uncom 3408	. 2 ⊢ (C ∪ A) = (A ∪ C)
3		uncom 3408	. 2 ⊢ (C ∪ B) = (B ∪ C)
4	1, 2, 3	3sstr4g 3312	1 ⊢ (A ⊆ B → (C ∪ A) ⊆ (C ∪ B))

Syntax hints:   → wi 4   ∪ cun 3207   ⊆ wss 3257

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  unss12  3435