Theorem i3orlem4 537

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem4	((a ∪ c)⊥ ∩ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem4

Step	Hyp	Ref	Expression
1		leo 150	. . 3 ((a ∪ c)⊥ ∩ (b ∪ c)) ≤ (((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ))
2	1	ler 141	. 2 ((a ∪ c)⊥ ∩ (b ∪ c)) ≤ ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
3		df-i3 45	. . 3 ((a ∪ c) →3 (b ∪ c)) = ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
4	3	ax-r1 34	. 2 ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))) = ((a ∪ c) →3 (b ∪ c))
5	2, 4	lbtr 131	1 ((a ∪ c)⊥ ∩ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i3 45  df-le1 122  df-le2 123

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