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Theorem womle2a 295
 Description: An equivalent to the WOM law.
Hypothesis
Ref Expression
womle2a.1 (a ∩ (a2 b)) ≤ ((a2 b) ∪ (a1 b))
Assertion
Ref Expression
womle2a ((a2 b) ∪ (a1 b)) = 1

Proof of Theorem womle2a
StepHypRef Expression
1 or4 84 . . 3 (((a2 b) ∪ (a2 b) ) ∪ ((a1 b) ∪ a )) = (((a2 b) ∪ (a1 b)) ∪ ((a2 b)a ))
2 oridm 110 . . . 4 ((a2 b) ∪ (a2 b) ) = (a2 b)
3 df-i1 44 . . . . . 6 (a1 b) = (a ∪ (ab))
43ax-r5 38 . . . . 5 ((a1 b) ∪ a ) = ((a ∪ (ab)) ∪ a )
5 oridm 110 . . . . . . 7 (aa ) = a
65ax-r5 38 . . . . . 6 ((aa ) ∪ (ab)) = (a ∪ (ab))
7 or32 82 . . . . . 6 ((a ∪ (ab)) ∪ a ) = ((aa ) ∪ (ab))
86, 7, 33tr1 63 . . . . 5 ((a ∪ (ab)) ∪ a ) = (a1 b)
94, 8ax-r2 36 . . . 4 ((a1 b) ∪ a ) = (a1 b)
102, 92or 72 . . 3 (((a2 b) ∪ (a2 b) ) ∪ ((a1 b) ∪ a )) = ((a2 b) ∪ (a1 b))
11 ax-a2 31 . . . . 5 ((a2 b)a ) = (a ∪ (a2 b) )
12 oran3 93 . . . . 5 (a ∪ (a2 b) ) = (a ∩ (a2 b))
1311, 12ax-r2 36 . . . 4 ((a2 b)a ) = (a ∩ (a2 b))
1413lor 70 . . 3 (((a2 b) ∪ (a1 b)) ∪ ((a2 b)a )) = (((a2 b) ∪ (a1 b)) ∪ (a ∩ (a2 b)) )
151, 10, 143tr2 64 . 2 ((a2 b) ∪ (a1 b)) = (((a2 b) ∪ (a1 b)) ∪ (a ∩ (a2 b)) )
16 le1 146 . . 3 (((a2 b) ∪ (a1 b)) ∪ (a ∩ (a2 b)) ) ≤ 1
17 df-t 41 . . . 4 1 = ((a ∩ (a2 b)) ∪ (a ∩ (a2 b)) )
18 womle2a.1 . . . . 5 (a ∩ (a2 b)) ≤ ((a2 b) ∪ (a1 b))
1918leror 152 . . . 4 ((a ∩ (a2 b)) ∪ (a ∩ (a2 b)) ) ≤ (((a2 b) ∪ (a1 b)) ∪ (a ∩ (a2 b)) )
2017, 19bltr 138 . . 3 1 ≤ (((a2 b) ∪ (a1 b)) ∪ (a ∩ (a2 b)) )
2116, 20lebi 145 . 2 (((a2 b) ∪ (a1 b)) ∪ (a ∩ (a2 b)) ) = 1
2215, 21ax-r2 36 1 ((a2 b) ∪ (a1 b)) = 1
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   →2 wi2 13 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131 This theorem is referenced by:  womle  298
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