Theorem i3orlem1 534

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem1	((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem1

Step	Hyp	Ref	Expression
1		leor 151	. 2 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
2		df-i3 45	. . 3 ((a ∪ c) →3 (b ∪ c)) = ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
3	2	ax-r1 34	. 2 ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))) = ((a ∪ c) →3 (b ∪ c))
4	1, 3	lbtr 131	1 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))

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

This theorem is referenced by:  i3orlem2 535  i3orlem3 536

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  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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