Theorem unssi 2208

Description: An inference that 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 285	. 2 ⊢ (A ⊆ C ⋀ B ⊆ C)
4		unss 2207	. 2 ⊢ ((A ⊆ C ⋀ B ⊆ C) ↔ (A ∪ B) ⊆ C)
5	3, 4	mpbi 189	1 ⊢ (A ∪ B) ⊆ C

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

This theorem is referenced by:  dmrnssfld 3363  rankun 4701  nn0ssre 6105  nn0ssz 6154  shslej 9333  shlub 9341  shsumval3 9356  shjshs 9410  spanun 9462  sshhococ 9464  osum 9581  cdrci 10480

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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