Theorem 3oalem4 31870

Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
3oalem4.3	⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴))

Assertion

Ref	Expression
3oalem4	⊢ 𝑅 ⊆ (⊥‘𝐵)

Proof of Theorem 3oalem4

Step	Hyp	Ref	Expression
1		3oalem4.3	. 2 ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴))
2		inss1 4190	. 2 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ⊆ (⊥‘𝐵)
3	1, 2	eqsstri 3984	1 ⊢ 𝑅 ⊆ (⊥‘𝐵)

Syntax hints:   = wceq 1562   ∩ cin 3905   ⊆ wss 3906  ‘cfv 6523  (class class class)co 7398  ⊥cort 31135   ∨_ℋ chj 31138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-in 3913  df-ss 3923

This theorem is referenced by:  3oalem5  31871