Theorem unssi 3404

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 3403	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
5	3, 4	mpbi 145	1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶

Syntax hints:   ∧ wa 104   ∪ cun 3218   ⊆ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by:  undifabs  3601  inundifss  3602  dmrnssfld  5040  caserel  7417  ltrelxr  8376  nn0ssre  9546  nn0ssz  9641  strleun  13435