Theorem bnj931 35401

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 4124	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		bnj931.1	. 2 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
3	1, 2	sseqtrri 3980	1 ⊢ 𝐵 ⊆ 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3897   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916

This theorem is used by:  bnj945  35404  bnj545  35525  bnj548  35527  bnj570  35535  bnj929  35566  bnj1136  35627  bnj1408  35666  bnj1442  35679