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Theorem wlem1 243
 Description: Lemma for 2-variable WOML proof. (Contributed by NM, 11-Nov-1998.)
Assertion
Ref Expression
wlem1 ((ab) ∪ ((a1 b) ∩ (b1 a))) = 1

Proof of Theorem wlem1
StepHypRef Expression
1 le1 146 . 2 ((ab) ∪ ((a1 b) ∩ (b1 a))) ≤ 1
2 df-t 41 . . . 4 1 = ((ab) ∪ (ab) )
3 ax-a2 31 . . . 4 ((ab) ∪ (ab) ) = ((ab) ∪ (ab))
42, 3ax-r2 36 . . 3 1 = ((ab) ∪ (ab))
5 dfb 94 . . . . 5 (ab) = ((ab) ∪ (ab ))
6 ledio 176 . . . . . 6 ((ab) ∪ (ab )) ≤ (((ab) ∪ a ) ∩ ((ab) ∪ b ))
7 df-i1 44 . . . . . . . . 9 (a1 b) = (a ∪ (ab))
8 ax-a2 31 . . . . . . . . 9 (a ∪ (ab)) = ((ab) ∪ a )
97, 8ax-r2 36 . . . . . . . 8 (a1 b) = ((ab) ∪ a )
10 df-i1 44 . . . . . . . . 9 (b1 a) = (b ∪ (ba))
11 ax-a2 31 . . . . . . . . . 10 (b ∪ (ba)) = ((ba) ∪ b )
12 ancom 74 . . . . . . . . . . 11 (ba) = (ab)
1312ax-r5 38 . . . . . . . . . 10 ((ba) ∪ b ) = ((ab) ∪ b )
1411, 13ax-r2 36 . . . . . . . . 9 (b ∪ (ba)) = ((ab) ∪ b )
1510, 14ax-r2 36 . . . . . . . 8 (b1 a) = ((ab) ∪ b )
169, 152an 79 . . . . . . 7 ((a1 b) ∩ (b1 a)) = (((ab) ∪ a ) ∩ ((ab) ∪ b ))
1716ax-r1 35 . . . . . 6 (((ab) ∪ a ) ∩ ((ab) ∪ b )) = ((a1 b) ∩ (b1 a))
186, 17lbtr 139 . . . . 5 ((ab) ∪ (ab )) ≤ ((a1 b) ∩ (b1 a))
195, 18bltr 138 . . . 4 (ab) ≤ ((a1 b) ∩ (b1 a))
2019lelor 166 . . 3 ((ab) ∪ (ab)) ≤ ((ab) ∪ ((a1 b) ∩ (b1 a)))
214, 20bltr 138 . 2 1 ≤ ((ab) ∪ ((a1 b) ∩ (b1 a)))
221, 21lebi 145 1 ((ab) ∪ ((a1 b) ∩ (b1 a))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →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 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 This theorem is referenced by:  wr5-2v  366
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