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Theorem u1lemle2 715
 Description: Sasaki implication to l.e.
Hypothesis
Ref Expression
u1lemle2.1 (a1 b) = 1
Assertion
Ref Expression
u1lemle2 ab

Proof of Theorem u1lemle2
StepHypRef Expression
1 anidm 111 . . . . . . . . 9 (aa) = a
21ran 78 . . . . . . . 8 ((aa) ∩ b) = (ab)
32ax-r1 35 . . . . . . 7 (ab) = ((aa) ∩ b)
4 anass 76 . . . . . . 7 ((aa) ∩ b) = (a ∩ (ab))
53, 4ax-r2 36 . . . . . 6 (ab) = (a ∩ (ab))
6 dff 101 . . . . . 6 0 = (aa )
75, 62or 72 . . . . 5 ((ab) ∪ 0) = ((a ∩ (ab)) ∪ (aa ))
8 ax-a2 31 . . . . . . . 8 (a ∪ (ab)) = ((ab) ∪ a )
98lan 77 . . . . . . 7 (a ∩ (a ∪ (ab))) = (a ∩ ((ab) ∪ a ))
10 coman1 185 . . . . . . . 8 (ab) C a
1110comcom2 183 . . . . . . . 8 (ab) C a
1210, 11fh2 470 . . . . . . 7 (a ∩ ((ab) ∪ a )) = ((a ∩ (ab)) ∪ (aa ))
139, 12ax-r2 36 . . . . . 6 (a ∩ (a ∪ (ab))) = ((a ∩ (ab)) ∪ (aa ))
1413ax-r1 35 . . . . 5 ((a ∩ (ab)) ∪ (aa )) = (a ∩ (a ∪ (ab)))
157, 14ax-r2 36 . . . 4 ((ab) ∪ 0) = (a ∩ (a ∪ (ab)))
16 df-i1 44 . . . . . . 7 (a1 b) = (a ∪ (ab))
1716ax-r1 35 . . . . . 6 (a ∪ (ab)) = (a1 b)
18 u1lemle2.1 . . . . . 6 (a1 b) = 1
1917, 18ax-r2 36 . . . . 5 (a ∪ (ab)) = 1
2019lan 77 . . . 4 (a ∩ (a ∪ (ab))) = (a ∩ 1)
2115, 20ax-r2 36 . . 3 ((ab) ∪ 0) = (a ∩ 1)
22 or0 102 . . 3 ((ab) ∪ 0) = (ab)
23 an1 106 . . 3 (a ∩ 1) = a
2421, 22, 233tr2 64 . 2 (ab) = a
2524df2le1 135 1 ab
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  3vded11  814  3vded12  815
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