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Theorem u2lemle2 716
 Description: Dishkant implication to l.e.
Hypothesis
Ref Expression
u2lemle2.1 (a2 b) = 1
Assertion
Ref Expression
u2lemle2 ab

Proof of Theorem u2lemle2
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (b ∪ (ab )) = ((ab ) ∪ b)
21lan 77 . . . . . 6 (a ∩ (b ∪ (ab ))) = (a ∩ ((ab ) ∪ b))
3 coman1 185 . . . . . . . . 9 (ab ) C a
43comcom7 460 . . . . . . . 8 (ab ) C a
5 coman2 186 . . . . . . . . 9 (ab ) C b
65comcom7 460 . . . . . . . 8 (ab ) C b
74, 6fh2 470 . . . . . . 7 (a ∩ ((ab ) ∪ b)) = ((a ∩ (ab )) ∪ (ab))
8 ancom 74 . . . . . . . . . 10 ((aa ) ∩ b ) = (b ∩ (aa ))
9 anass 76 . . . . . . . . . 10 ((aa ) ∩ b ) = (a ∩ (ab ))
10 dff 101 . . . . . . . . . . . . 13 0 = (aa )
1110ax-r1 35 . . . . . . . . . . . 12 (aa ) = 0
1211lan 77 . . . . . . . . . . 11 (b ∩ (aa )) = (b ∩ 0)
13 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
1412, 13ax-r2 36 . . . . . . . . . 10 (b ∩ (aa )) = 0
158, 9, 143tr2 64 . . . . . . . . 9 (a ∩ (ab )) = 0
1615ax-r5 38 . . . . . . . 8 ((a ∩ (ab )) ∪ (ab)) = (0 ∪ (ab))
17 ax-a2 31 . . . . . . . 8 (0 ∪ (ab)) = ((ab) ∪ 0)
1816, 17ax-r2 36 . . . . . . 7 ((a ∩ (ab )) ∪ (ab)) = ((ab) ∪ 0)
197, 18ax-r2 36 . . . . . 6 (a ∩ ((ab ) ∪ b)) = ((ab) ∪ 0)
202, 19ax-r2 36 . . . . 5 (a ∩ (b ∪ (ab ))) = ((ab) ∪ 0)
2120ax-r1 35 . . . 4 ((ab) ∪ 0) = (a ∩ (b ∪ (ab )))
22 df-i2 45 . . . . . . 7 (a2 b) = (b ∪ (ab ))
2322ax-r1 35 . . . . . 6 (b ∪ (ab )) = (a2 b)
24 u2lemle2.1 . . . . . 6 (a2 b) = 1
2523, 24ax-r2 36 . . . . 5 (b ∪ (ab )) = 1
2625lan 77 . . . 4 (a ∩ (b ∪ (ab ))) = (a ∩ 1)
2721, 26ax-r2 36 . . 3 ((ab) ∪ 0) = (a ∩ 1)
28 or0 102 . . 3 ((ab) ∪ 0) = (ab)
29 an1 106 . . 3 (a ∩ 1) = a
3027, 28, 293tr2 64 . 2 (ab) = a
3130df2le1 135 1 ab
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →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  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  3vroa  831  imp3  841
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