Theorem unssi 3347

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 272	. 2 ⊢ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)
4		unss 3346	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
5	3, 4	mpbi 145	1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶

Syntax hints:   ∧ wa 104   ∪ cun 3163   ⊆ wss 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178

This theorem is referenced by:  undifabs  3536  inundifss  3537  dmrnssfld  4940  caserel  7188  ltrelxr  8132  nn0ssre  9298  nn0ssz  9389  strleun  12907