Theorem unssi 3348

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 3347	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
5	3, 4	mpbi 145	1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶

Syntax hints:   ∧ wa 104   ∪ cun 3164   ⊆ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179

This theorem is referenced by:  undifabs  3537  inundifss  3538  dmrnssfld  4941  caserel  7189  ltrelxr  8133  nn0ssre  9299  nn0ssz  9390  strleun  12936