Theorem i3orlem3 554

Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.)

Assertion

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

Proof of Theorem i3orlem3

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . . . 6 ((a ∪ c)⊥ ∪ c) = (c ∪ (a ∪ c)⊥ )
2	1	lan 77	. . . . 5 (c ∩ ((a ∪ c)⊥ ∪ c)) = (c ∩ (c ∪ (a ∪ c)⊥ ))
3		anabs 121	. . . . 5 (c ∩ (c ∪ (a ∪ c)⊥ )) = c
4	2, 3	ax-r2 36	. . . 4 (c ∩ ((a ∪ c)⊥ ∪ c)) = c
5	4	ax-r1 35	. . 3 c = (c ∩ ((a ∪ c)⊥ ∪ c))
6		leor 159	. . . 4 c ≤ (a ∪ c)
7		leor 159	. . . . 5 c ≤ (b ∪ c)
8	7	lelor 166	. . . 4 ((a ∪ c)⊥ ∪ c) ≤ ((a ∪ c)⊥ ∪ (b ∪ c))
9	6, 8	le2an 169	. . 3 (c ∩ ((a ∪ c)⊥ ∪ c)) ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))
10	5, 9	bltr 138	. 2 c ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))
11		i3orlem1 552	. 2 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))
12	10, 11	letr 137	1 c ≤ ((a ∪ c) →3 (b ∪ c))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131

This theorem is referenced by: (None)