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Theorem gt1 492
 Description: Part of Lemma 1 from Gaisi Takeuti, "Quantum Set Theory". (Contributed by NM, 2-Dec-1998.)
Hypotheses
Ref Expression
gt1.1 a = (bc)
gt1.2 bd
gt1.3 cd
Assertion
Ref Expression
gt1 a C d

Proof of Theorem gt1
StepHypRef Expression
1 gt1.1 . 2 a = (bc)
2 gt1.2 . . . . . 6 bd
32lecom 180 . . . . 5 b C d
43comcom 453 . . . 4 d C b
5 gt1.3 . . . . . . 7 cd
65lecom 180 . . . . . 6 c C d
76comcom7 460 . . . . 5 c C d
87comcom 453 . . . 4 d C c
94, 8com2or 483 . . 3 d C (bc)
109comcom 453 . 2 (bc) C d
111, 10bctr 181 1 a C d
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   C wc 3  ⊥ wn 4   ∪ wo 6 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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