Theorem 3oalem4 31756

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 4167	. 2 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ⊆ (⊥‘𝐵)
3	1, 2	eqsstri 3962	1 ⊢ 𝑅 ⊆ (⊥‘𝐵)

Syntax hints:   = wceq 1548   ∩ cin 3883   ⊆ wss 3884  ‘cfv 6488  (class class class)co 7359  ⊥cort 31021   ∨_ℋ chj 31024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-in 3891  df-ss 3901

This theorem is referenced by:  3oalem5  31757