Theorem bnj931 34236

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj931.1	⊢ 𝐴 = (𝐵 ∪ 𝐶)

Assertion

Ref	Expression
bnj931	⊢ 𝐵 ⊆ 𝐴

Proof of Theorem bnj931

Step	Hyp	Ref	Expression
1		ssun1 4164	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		bnj931.1	. 2 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
3	1, 2	sseqtrri 4011	1 ⊢ 𝐵 ⊆ 𝐴

Syntax hints:   = wceq 1533   ∪ cun 3938   ⊆ wss 3940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3945  df-in 3947  df-ss 3957

This theorem is referenced by:  bnj945  34239  bnj545  34361  bnj548  34363  bnj570  34371  bnj929  34402  bnj1136  34463  bnj1408  34502  bnj1442  34515