Theorem negantlem4 833

Description: Lemma for negated antecedent identity.

Hypothesis

Ref	Expression
negant.1	(a →1 c) = (b →1 c)

Assertion

Ref	Expression
negantlem4	(a⊥ →1 c) ≤ (b⊥ →1 c)

Proof of Theorem negantlem4

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
2		ax-a1 29	. . . . 5 a = a⊥ ⊥
3	2	ax-r5 37	. . . 4 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
4	3	ax-r1 34	. . 3 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a ∪ (a⊥ ∩ c))
5	1, 4	ax-r2 35	. 2 (a⊥ →1 c) = (a ∪ (a⊥ ∩ c))
6		negant.1	. . . 4 (a →1 c) = (b →1 c)
7	6	negantlem2 831	. . 3 a ≤ (b⊥ →1 c)
8	6	negantlem3 832	. . 3 (a⊥ ∩ c) ≤ (b⊥ →1 c)
9	7, 8	lel2or 162	. 2 (a ∪ (a⊥ ∩ c)) ≤ (b⊥ →1 c)
10	5, 9	bltr 130	1 (a⊥ →1 c) ≤ (b⊥ →1 c)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  negant 834

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org