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Theorem test2 803
Description: Part of an attempt to crack a potential Kalmbach axiom. (Contributed by NM, 29-Dec-1997.)
Assertion
Ref Expression
test2 (ab) ≤ ((ab) ∪ ((c ∪ (ab)) ∩ (c ∪ (ab))))

Proof of Theorem test2
StepHypRef Expression
1 dfnb 95 . . . . 5 (ab) = ((ab) ∩ (ab ))
2 anidm 111 . . . . 5 ((ab) ∩ (ab)) = (ab)
31, 22or 72 . . . 4 ((ab) ∪ ((ab) ∩ (ab))) = (((ab) ∩ (ab )) ∪ (ab))
4 comor1 461 . . . . . . 7 (ab) C a
5 comor2 462 . . . . . . 7 (ab) C b
64, 5com2an 484 . . . . . 6 (ab) C (ab)
74comcom2 183 . . . . . . 7 (ab) C a
85comcom2 183 . . . . . . 7 (ab) C b
97, 8com2or 483 . . . . . 6 (ab) C (ab )
106, 9fh4r 476 . . . . 5 (((ab) ∩ (ab )) ∪ (ab)) = (((ab) ∪ (ab)) ∩ ((ab ) ∪ (ab)))
11 ax-a2 31 . . . . . . . 8 ((ab) ∪ (ab)) = ((ab) ∪ (ab))
12 lea 160 . . . . . . . . . 10 (ab) ≤ a
13 leo 158 . . . . . . . . . 10 a ≤ (ab)
1412, 13letr 137 . . . . . . . . 9 (ab) ≤ (ab)
1514df-le2 131 . . . . . . . 8 ((ab) ∪ (ab)) = (ab)
1611, 15ax-r2 36 . . . . . . 7 ((ab) ∪ (ab)) = (ab)
17 df-a 40 . . . . . . . . 9 (ab) = (ab )
1817lor 70 . . . . . . . 8 ((ab ) ∪ (ab)) = ((ab ) ∪ (ab ) )
19 df-t 41 . . . . . . . . 9 1 = ((ab ) ∪ (ab ) )
2019ax-r1 35 . . . . . . . 8 ((ab ) ∪ (ab ) ) = 1
2118, 20ax-r2 36 . . . . . . 7 ((ab ) ∪ (ab)) = 1
2216, 212an 79 . . . . . 6 (((ab) ∪ (ab)) ∩ ((ab ) ∪ (ab))) = ((ab) ∩ 1)
23 an1 106 . . . . . 6 ((ab) ∩ 1) = (ab)
2422, 23ax-r2 36 . . . . 5 (((ab) ∪ (ab)) ∩ ((ab ) ∪ (ab))) = (ab)
2510, 24ax-r2 36 . . . 4 (((ab) ∩ (ab )) ∪ (ab)) = (ab)
263, 25ax-r2 36 . . 3 ((ab) ∪ ((ab) ∩ (ab))) = (ab)
2726ax-r1 35 . 2 (ab) = ((ab) ∪ ((ab) ∩ (ab)))
28 leor 159 . . . 4 (ab) ≤ (c ∪ (ab))
29 leor 159 . . . 4 (ab) ≤ (c ∪ (ab))
3028, 29le2an 169 . . 3 ((ab) ∩ (ab)) ≤ ((c ∪ (ab)) ∩ (c ∪ (ab)))
3130lelor 166 . 2 ((ab) ∪ ((ab) ∩ (ab))) ≤ ((ab) ∪ ((c ∪ (ab)) ∩ (c ∪ (ab))))
3227, 31bltr 138 1 (ab) ≤ ((ab) ∪ ((c ∪ (ab)) ∩ (c ∪ (ab))))
Colors of variables: term
Syntax hints:  wle 2   wn 4  tb 5  wo 6  wa 7  1wt 8
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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