Theorem bnj931 30095

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 3738	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		bnj931.1	. 2 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
3	1, 2	sseqtr4i 3601	1 ⊢ 𝐵 ⊆ 𝐴

Syntax hints:   = wceq 1475   ∪ cun 3538   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554

This theorem is referenced by:  bnj945  30098  bnj545  30219  bnj548  30221  bnj570  30229  bnj929  30260  bnj1136  30319  bnj1408  30358  bnj1442  30371