Theorem 3oalem4 32132

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 4185	. 2 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ⊆ (⊥‘𝐵)
3	1, 2	eqsstri 3980	1 ⊢ 𝑅 ⊆ (⊥‘𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3901   ⊆ wss 3902  ‘cfv 6537  (class class class)co 7416  ⊥cort 31397   ∨_ℋ chj 31400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-ss 3919

This theorem is used by:  3oalem5  32133