Theorem unss12 2205

Description: Subclass law for union of classes.

Assertion

Ref	Expression
unss12	⊢ ((A ⊆ B ⋀ C ⊆ D) → (A ∪ C) ⊆ (B ∪ D))

Proof of Theorem unss12

Step	Hyp	Ref	Expression
1		unss1 2202	. 2 ⊢ (A ⊆ B → (A ∪ C) ⊆ (B ∪ C))
2		unss2 2204	. 2 ⊢ (C ⊆ D → (B ∪ C) ⊆ (B ∪ D))
3	1, 2	sylan9ss 2078	1 ⊢ ((A ⊆ B ⋀ C ⊆ D) → (A ∪ C) ⊆ (B ∪ D))

Syntax hints:   → wi 3   ⋀ wa 223   ∪ cun 2048   ⊆ wss 2050

This theorem is referenced by:  pwssun 2833  fun 3647  undom 4444  spanun 9462  sshhococ 9464

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056

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