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Theorem u4lemle2 718
 Description: Non-tollens implication to l.e. (Contributed by NM, 11-Jan-1998.)
Hypothesis
Ref Expression
u4lemle2.1 (a4 b) = 1
Assertion
Ref Expression
u4lemle2 ab

Proof of Theorem u4lemle2
StepHypRef Expression
1 df-i4 47 . . . . . 6 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
21ax-r1 35 . . . . 5 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) = (a4 b)
3 u4lemle2.1 . . . . 5 (a4 b) = 1
42, 3ax-r2 36 . . . 4 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) = 1
54lan 77 . . 3 (a ∩ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = (a ∩ 1)
6 comanr1 464 . . . . . . 7 a C (ab)
7 comanr1 464 . . . . . . . 8 a C (ab)
87comcom6 459 . . . . . . 7 a C (ab)
96, 8com2or 483 . . . . . 6 a C ((ab) ∪ (ab))
109comcom 453 . . . . 5 ((ab) ∪ (ab)) C a
11 comor1 461 . . . . . . . . . 10 (ab) C a
1211comcom7 460 . . . . . . . . 9 (ab) C a
13 comor2 462 . . . . . . . . 9 (ab) C b
1412, 13com2an 484 . . . . . . . 8 (ab) C (ab)
1511, 13com2an 484 . . . . . . . 8 (ab) C (ab)
1614, 15com2or 483 . . . . . . 7 (ab) C ((ab) ∪ (ab))
1716comcom 453 . . . . . 6 ((ab) ∪ (ab)) C (ab)
18 comanr2 465 . . . . . . . . 9 b C (ab)
1918comcom3 454 . . . . . . . 8 b C (ab)
20 comanr2 465 . . . . . . . . 9 b C (ab)
2120comcom3 454 . . . . . . . 8 b C (ab)
2219, 21com2or 483 . . . . . . 7 b C ((ab) ∪ (ab))
2322comcom 453 . . . . . 6 ((ab) ∪ (ab)) C b
2417, 23com2an 484 . . . . 5 ((ab) ∪ (ab)) C ((ab) ∩ b )
2510, 24fh2 470 . . . 4 (a ∩ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = ((a ∩ ((ab) ∪ (ab))) ∪ (a ∩ ((ab) ∩ b )))
266, 8fh1 469 . . . . . . 7 (a ∩ ((ab) ∪ (ab))) = ((a ∩ (ab)) ∪ (a ∩ (ab)))
27 anidm 111 . . . . . . . . . . . . 13 (aa) = a
2827ran 78 . . . . . . . . . . . 12 ((aa) ∩ b) = (ab)
2928ax-r1 35 . . . . . . . . . . 11 (ab) = ((aa) ∩ b)
30 anass 76 . . . . . . . . . . 11 ((aa) ∩ b) = (a ∩ (ab))
3129, 30ax-r2 36 . . . . . . . . . 10 (ab) = (a ∩ (ab))
32 dff 101 . . . . . . . . . . . . 13 0 = (aa )
3332lan 77 . . . . . . . . . . . 12 (b ∩ 0) = (b ∩ (aa ))
34 an0 108 . . . . . . . . . . . 12 (b ∩ 0) = 0
35 ancom 74 . . . . . . . . . . . 12 (b ∩ (aa )) = ((aa ) ∩ b)
3633, 34, 353tr2 64 . . . . . . . . . . 11 0 = ((aa ) ∩ b)
37 anass 76 . . . . . . . . . . 11 ((aa ) ∩ b) = (a ∩ (ab))
3836, 37ax-r2 36 . . . . . . . . . 10 0 = (a ∩ (ab))
3931, 382or 72 . . . . . . . . 9 ((ab) ∪ 0) = ((a ∩ (ab)) ∪ (a ∩ (ab)))
4039ax-r1 35 . . . . . . . 8 ((a ∩ (ab)) ∪ (a ∩ (ab))) = ((ab) ∪ 0)
41 or0 102 . . . . . . . 8 ((ab) ∪ 0) = (ab)
4240, 41ax-r2 36 . . . . . . 7 ((a ∩ (ab)) ∪ (a ∩ (ab))) = (ab)
4326, 42ax-r2 36 . . . . . 6 (a ∩ ((ab) ∪ (ab))) = (ab)
44 anor1 88 . . . . . . . 8 (ab ) = (ab)
4544lan 77 . . . . . . 7 ((ab) ∩ (ab )) = ((ab) ∩ (ab) )
46 an12 81 . . . . . . 7 (a ∩ ((ab) ∩ b )) = ((ab) ∩ (ab ))
47 dff 101 . . . . . . 7 0 = ((ab) ∩ (ab) )
4845, 46, 473tr1 63 . . . . . 6 (a ∩ ((ab) ∩ b )) = 0
4943, 482or 72 . . . . 5 ((a ∩ ((ab) ∪ (ab))) ∪ (a ∩ ((ab) ∩ b ))) = ((ab) ∪ 0)
5049, 41ax-r2 36 . . . 4 ((a ∩ ((ab) ∪ (ab))) ∪ (a ∩ ((ab) ∩ b ))) = (ab)
5125, 50ax-r2 36 . . 3 (a ∩ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = (ab)
52 an1 106 . . 3 (a ∩ 1) = a
535, 51, 523tr2 64 . 2 (ab) = a
5453df2le1 135 1 ab
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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