Theorem bnj931 35159

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 4131	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		bnj931.1	. 2 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
3	1, 2	sseqtrri 3986	1 ⊢ 𝐵 ⊆ 𝐴

Syntax hints:   = wceq 1570   ∪ cun 3903   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3910  df-ss 3922

This theorem is referenced by:  bnj945  35162  bnj545  35283  bnj548  35285  bnj570  35293  bnj929  35324  bnj1136  35385  bnj1408  35424  bnj1442  35437