Theorem 3oalem4 31646

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 4212	. 2 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ⊆ (⊥‘𝐵)
3	1, 2	eqsstri 4005	1 ⊢ 𝑅 ⊆ (⊥‘𝐵)

Syntax hints:   = wceq 1540   ∩ cin 3925   ⊆ wss 3926  ‘cfv 6531  (class class class)co 7405  ⊥cort 30911   ∨_ℋ chj 30914

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-in 3933  df-ss 3943

This theorem is referenced by:  3oalem5  31647