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Theorem oa4to4u2 974
 Description: A weaker-looking "universal" proper 4-OA.
Hypotheses
Ref Expression
oa4to4u.1 ((e1 d) ∩ (e ∪ (f ∩ (((ef) ∪ ((e1 d) ∩ (f1 d))) ∪ (((eg) ∪ ((e1 d) ∩ (g1 d))) ∩ ((fg) ∪ ((f1 d) ∩ (g1 d)))))))) ≤ (((ed) ∪ (fd)) ∪ (gd))
oa4to4u.2 e = (a1 d)
oa4to4u3 f = (b1 d)
oa4to4u.4 g = (c1 d)
Assertion
Ref Expression
oa4to4u2 ((a1 d) ∩ ((a1 d) ∪ ((b1 d) ∩ ((((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d))) ∪ ((((a1 d) ∩ (c1 d)) ∪ ((a1 d) ∩ (c1 d))) ∩ (((b1 d) ∩ (c1 d)) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ d

Proof of Theorem oa4to4u2
StepHypRef Expression
1 oa4to4u.1 . . 3 ((e1 d) ∩ (e ∪ (f ∩ (((ef) ∪ ((e1 d) ∩ (f1 d))) ∪ (((eg) ∪ ((e1 d) ∩ (g1 d))) ∩ ((fg) ∪ ((f1 d) ∩ (g1 d)))))))) ≤ (((ed) ∪ (fd)) ∪ (gd))
2 oa4to4u.2 . . 3 e = (a1 d)
3 oa4to4u3 . . 3 f = (b1 d)
4 oa4to4u.4 . . 3 g = (c1 d)
51, 2, 3, 4oa4to4u 973 . 2 ((a1 d) ∩ ((a1 d) ∪ ((b1 d) ∩ ((((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d))) ∪ ((((a1 d) ∩ (c1 d)) ∪ ((a1 d) ∩ (c1 d))) ∩ (((b1 d) ∩ (c1 d)) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ ((((a1 d) ∩ (a1 d)) ∪ ((b1 d) ∩ (b1 d))) ∪ ((c1 d) ∩ (c1 d)))
6 u1lem8 776 . . . . 5 ((a1 d) ∩ (a1 d)) = ((ad) ∪ (ad))
7 lear 161 . . . . . 6 (ad) ≤ d
8 lear 161 . . . . . 6 (ad) ≤ d
97, 8lel2or 170 . . . . 5 ((ad) ∪ (ad)) ≤ d
106, 9bltr 138 . . . 4 ((a1 d) ∩ (a1 d)) ≤ d
11 u1lem8 776 . . . . 5 ((b1 d) ∩ (b1 d)) = ((bd) ∪ (bd))
12 lear 161 . . . . . 6 (bd) ≤ d
13 lear 161 . . . . . 6 (bd) ≤ d
1412, 13lel2or 170 . . . . 5 ((bd) ∪ (bd)) ≤ d
1511, 14bltr 138 . . . 4 ((b1 d) ∩ (b1 d)) ≤ d
1610, 15lel2or 170 . . 3 (((a1 d) ∩ (a1 d)) ∪ ((b1 d) ∩ (b1 d))) ≤ d
17 u1lem8 776 . . . 4 ((c1 d) ∩ (c1 d)) = ((cd) ∪ (cd))
18 lear 161 . . . . 5 (cd) ≤ d
19 lear 161 . . . . 5 (cd) ≤ d
2018, 19lel2or 170 . . . 4 ((cd) ∪ (cd)) ≤ d
2117, 20bltr 138 . . 3 ((c1 d) ∩ (c1 d)) ≤ d
2216, 21lel2or 170 . 2 ((((a1 d) ∩ (a1 d)) ∪ ((b1 d) ∩ (b1 d))) ∪ ((c1 d) ∩ (c1 d))) ≤ d
235, 22letr 137 1 ((a1 d) ∩ ((a1 d) ∪ ((b1 d) ∩ ((((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d))) ∪ ((((a1 d) ∩ (c1 d)) ∪ ((a1 d) ∩ (c1 d))) ∩ (((b1 d) ∩ (c1 d)) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ d
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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  ax-r3 439 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  df-c1 132  df-c2 133 This theorem is referenced by:  4oa  1039
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