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Theorem u5lemle2 719
 Description: Relevance implication to l.e. (Contributed by NM, 11-Jan-1998.)
Hypothesis
Ref Expression
u5lemle2.1 (a5 b) = 1
Assertion
Ref Expression
u5lemle2 ab

Proof of Theorem u5lemle2
StepHypRef Expression
1 df-i5 48 . . . . . 6 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
21ax-r1 35 . . . . 5 (((ab) ∪ (ab)) ∪ (ab )) = (a5 b)
3 u5lemle2.1 . . . . 5 (a5 b) = 1
42, 3ax-r2 36 . . . 4 (((ab) ∪ (ab)) ∪ (ab )) = 1
54lan 77 . . 3 (a ∩ (((ab) ∪ (ab)) ∪ (ab ))) = (a ∩ 1)
6 comanr1 464 . . . . . 6 a C (ab)
7 comanr1 464 . . . . . . 7 a C (ab)
87comcom6 459 . . . . . 6 a C (ab)
96, 8com2or 483 . . . . 5 a C ((ab) ∪ (ab))
10 comanr1 464 . . . . . 6 a C (ab )
1110comcom6 459 . . . . 5 a C (ab )
129, 11fh1 469 . . . 4 (a ∩ (((ab) ∪ (ab)) ∪ (ab ))) = ((a ∩ ((ab) ∪ (ab))) ∪ (a ∩ (ab )))
136, 8fh1 469 . . . . . . 7 (a ∩ ((ab) ∪ (ab))) = ((a ∩ (ab)) ∪ (a ∩ (ab)))
14 anass 76 . . . . . . . . . . 11 ((aa) ∩ b) = (a ∩ (ab))
1514ax-r1 35 . . . . . . . . . 10 (a ∩ (ab)) = ((aa) ∩ b)
16 anidm 111 . . . . . . . . . . 11 (aa) = a
1716ran 78 . . . . . . . . . 10 ((aa) ∩ b) = (ab)
1815, 17ax-r2 36 . . . . . . . . 9 (a ∩ (ab)) = (ab)
19 ancom 74 . . . . . . . . . 10 ((aa ) ∩ b) = (b ∩ (aa ))
20 anass 76 . . . . . . . . . 10 ((aa ) ∩ b) = (a ∩ (ab))
21 dff 101 . . . . . . . . . . . . 13 0 = (aa )
2221ax-r1 35 . . . . . . . . . . . 12 (aa ) = 0
2322lan 77 . . . . . . . . . . 11 (b ∩ (aa )) = (b ∩ 0)
24 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
2523, 24ax-r2 36 . . . . . . . . . 10 (b ∩ (aa )) = 0
2619, 20, 253tr2 64 . . . . . . . . 9 (a ∩ (ab)) = 0
2718, 262or 72 . . . . . . . 8 ((a ∩ (ab)) ∪ (a ∩ (ab))) = ((ab) ∪ 0)
28 or0 102 . . . . . . . 8 ((ab) ∪ 0) = (ab)
2927, 28ax-r2 36 . . . . . . 7 ((a ∩ (ab)) ∪ (a ∩ (ab))) = (ab)
3013, 29ax-r2 36 . . . . . 6 (a ∩ ((ab) ∪ (ab))) = (ab)
31 ancom 74 . . . . . . 7 ((aa ) ∩ b ) = (b ∩ (aa ))
32 anass 76 . . . . . . 7 ((aa ) ∩ b ) = (a ∩ (ab ))
3321lan 77 . . . . . . . . 9 (b ∩ 0) = (b ∩ (aa ))
3433ax-r1 35 . . . . . . . 8 (b ∩ (aa )) = (b ∩ 0)
35 an0 108 . . . . . . . 8 (b ∩ 0) = 0
3634, 35ax-r2 36 . . . . . . 7 (b ∩ (aa )) = 0
3731, 32, 363tr2 64 . . . . . 6 (a ∩ (ab )) = 0
3830, 372or 72 . . . . 5 ((a ∩ ((ab) ∪ (ab))) ∪ (a ∩ (ab ))) = ((ab) ∪ 0)
3938, 28ax-r2 36 . . . 4 ((a ∩ ((ab) ∪ (ab))) ∪ (a ∩ (ab ))) = (ab)
4012, 39ax-r2 36 . . 3 (a ∩ (((ab) ∪ (ab)) ∪ (ab ))) = (ab)
41 an1 106 . . 3 (a ∩ 1) = a
425, 40, 413tr2 64 . 2 (ab) = a
4342df2le1 135 1 ab
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →5 wi5 16 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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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