Theorem i3orlem3 536

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem3	c ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem3

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . . 6 ((a ∪ c)⊥ ∪ c) = (c ∪ (a ∪ c)⊥ )
2	1	lan 70	. . . . 5 (c ∩ ((a ∪ c)⊥ ∪ c)) = (c ∩ (c ∪ (a ∪ c)⊥ ))
3		a5c 113	. . . . 5 (c ∩ (c ∪ (a ∪ c)⊥ )) = c
4	2, 3	ax-r2 35	. . . 4 (c ∩ ((a ∪ c)⊥ ∪ c)) = c
5	4	ax-r1 34	. . 3 c = (c ∩ ((a ∪ c)⊥ ∪ c))
6		leor 151	. . . 4 c ≤ (a ∪ c)
7		leor 151	. . . . 5 c ≤ (b ∪ c)
8	7	lelor 158	. . . 4 ((a ∪ c)⊥ ∪ c) ≤ ((a ∪ c)⊥ ∪ (b ∪ c))
9	6, 8	le2an 161	. . 3 (c ∩ ((a ∪ c)⊥ ∪ c)) ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))
10	5, 9	bltr 130	. 2 c ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))
11		i3orlem1 534	. 2 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))
12	10, 11	letr 129	1 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-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123

metamath.org