Theorem gstho 473

Description: "OR" version of Gudder-Schelp's Theorem.

Hypotheses

Ref	Expression
gstho.1	b C c
gstho.2	a C (b ∪ c)

Assertion

Ref	Expression
gstho	(a ∪ b) C c

Proof of Theorem gstho

Step	Hyp	Ref	Expression
1		anor3 82	. . . 4 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
2	1	ax-r1 34	. . 3 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
3		gstho.1	. . . . 5 b C c
4	3	comcom4 437	. . . 4 b⊥ C c⊥
5		gstho.2	. . . . . 6 a C (b ∪ c)
6	5	comcom4 437	. . . . 5 a⊥ C (b ∪ c)⊥
7		anor3 82	. . . . . 6 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
8	7	ax-r1 34	. . . . 5 (b ∪ c)⊥ = (b⊥ ∩ c⊥ )
9	6, 8	cbtr 174	. . . 4 a⊥ C (b⊥ ∩ c⊥ )
10	4, 9	gsth2 472	. . 3 (a⊥ ∩ b⊥ ) C c⊥
11	2, 10	bctr 173	. 2 (a ∪ b)⊥ C c⊥
12	11	comcom5 440	1 (a ∪ b) C c

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org