Theorem 3oalem4 29445

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 4208	. 2 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ⊆ (⊥‘𝐵)
3	1, 2	eqsstri 4004	1 ⊢ 𝑅 ⊆ (⊥‘𝐵)

Syntax hints:   = wceq 1536   ∩ cin 3938   ⊆ wss 3939  ‘cfv 6358  (class class class)co 7159  ⊥cort 28710   ∨_ℋ chj 28713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-in 3946  df-ss 3955

This theorem is referenced by:  3oalem5  29446