Theorem oadp35lemf 1210

Description: Part of proof (3)=>(5) in Day/Pickering 1982. (Contributed by NM, 12-Jul-2015.)

Hypotheses

Ref	Expression
oadp35lem.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
oadp35lem.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
oadp35lem.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
oadp35lem.4	p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
oadp35lem.5	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
oadp35lemf	(a0 ∪ p) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

Proof of Theorem oadp35lemf

Step	Hyp	Ref	Expression
1		leo 158	. 2 a0 ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
2		oadp35lem.1	. . 3 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
3		oadp35lem.2	. . 3 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
4		oadp35lem.3	. . 3 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
5		oadp35lem.4	. . 3 p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
6		oadp35lem.5	. . 3 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
7	2, 3, 4, 5, 6	oadp35lemg 1209	. 2 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
8	1, 7	lel2or 170	1 (a0 ∪ p) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1122  ax-arg 1153

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

This theorem is referenced by: (None)