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Theorem comi12 707
Description: Commutation theorem for 1 and 2 . (Contributed by NM, 5-Jul-2000.)
Assertion
Ref Expression
comi12 (a1 b) C (c2 a)

Proof of Theorem comi12
StepHypRef Expression
1 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2 lea 160 . . . . . . . 8 (a ∩ (ca ) ) ≤ a
3 leo 158 . . . . . . . 8 a ≤ (a ∪ (ab))
42, 3letr 137 . . . . . . 7 (a ∩ (ca ) ) ≤ (a ∪ (ab))
54lecom 180 . . . . . 6 (a ∩ (ca ) ) C (a ∪ (ab))
65comcom 453 . . . . 5 (a ∪ (ab)) C (a ∩ (ca ) )
7 anor3 90 . . . . 5 (a ∩ (ca ) ) = (a ∪ (ca ))
86, 7cbtr 182 . . . 4 (a ∪ (ab)) C (a ∪ (ca ))
98comcom7 460 . . 3 (a ∪ (ab)) C (a ∪ (ca ))
10 df-i2 45 . . . 4 (c2 a) = (a ∪ (ca ))
1110ax-r1 35 . . 3 (a ∪ (ca )) = (c2 a)
129, 11cbtr 182 . 2 (a ∪ (ab)) C (c2 a)
131, 12bctr 181 1 (a1 b) C (c2 a)
Colors of variables: term
Syntax hints:   C wc 3   wn 4  wo 6  wa 7  1 wi1 12  2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  orbi  842
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