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Theorem gsth2 490
 Description: Stronger version of Gudder-Schelp's Theorem. Beran, p. 263, Th. 4.2.
Hypotheses
Ref Expression
gsth2.1 b C c
gsth2.2 a C (bc)
Assertion
Ref Expression
gsth2 (ab) C c

Proof of Theorem gsth2
StepHypRef Expression
1 gsth2.1 . . . . 5 b C c
21comcom 453 . . . 4 c C b
3 ancom 74 . . . . . . . . 9 (b ∩ (ba )) = ((ba ) ∩ b)
4 ax-a2 31 . . . . . . . . . 10 (ba ) = (ab )
54ran 78 . . . . . . . . 9 ((ba ) ∩ b) = ((ab ) ∩ b)
63, 5ax-r2 36 . . . . . . . 8 (b ∩ (ba )) = ((ab ) ∩ b)
7 comor2 462 . . . . . . . . . 10 (ab ) C b
87comcom7 460 . . . . . . . . 9 (ab ) C b
9 gsth2.2 . . . . . . . . . . . . 13 a C (bc)
109comcom 453 . . . . . . . . . . . 12 (bc) C a
1110comcom2 183 . . . . . . . . . . 11 (bc) C a
12 coman1 185 . . . . . . . . . . . 12 (bc) C b
1312comcom2 183 . . . . . . . . . . 11 (bc) C b
1411, 13com2or 483 . . . . . . . . . 10 (bc) C (ab )
1514comcom 453 . . . . . . . . 9 (ab ) C (bc)
168, 1, 15gsth 489 . . . . . . . 8 ((ab ) ∩ b) C c
176, 16bctr 181 . . . . . . 7 (b ∩ (ba )) C c
1817comcom 453 . . . . . 6 c C (b ∩ (ba ))
19 df-a 40 . . . . . . 7 (b ∩ (ba )) = (b ∪ (ba ) )
20 df-a 40 . . . . . . . . . 10 (ba) = (ba )
2120lor 70 . . . . . . . . 9 (b ∪ (ba)) = (b ∪ (ba ) )
2221ax-r4 37 . . . . . . . 8 (b ∪ (ba)) = (b ∪ (ba ) )
2322ax-r1 35 . . . . . . 7 (b ∪ (ba ) ) = (b ∪ (ba))
2419, 23ax-r2 36 . . . . . 6 (b ∩ (ba )) = (b ∪ (ba))
2518, 24cbtr 182 . . . . 5 c C (b ∪ (ba))
2625comcom7 460 . . . 4 c C (b ∪ (ba))
272, 26com2an 484 . . 3 c C (b ∩ (b ∪ (ba)))
28 omla 447 . . . 4 (b ∩ (b ∪ (ba))) = (ba)
29 ancom 74 . . . 4 (ba) = (ab)
3028, 29ax-r2 36 . . 3 (b ∩ (b ∪ (ba))) = (ab)
3127, 30cbtr 182 . 2 c C (ab)
3231comcom 453 1 (ab) C c
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7 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:  gstho  491  oacom  1011  oacom3  1013
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