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Theorem i2or 344
 Description: Lemma for disjunction of →2 . (Contributed by NM, 5-Jul-2000.)
Assertion
Ref Expression
i2or ((a2 c) ∪ (b2 c)) ≤ ((ab) →2 c)

Proof of Theorem i2or
StepHypRef Expression
1 df-i2 45 . . . 4 (a2 c) = (c ∪ (ac ))
2 lea 160 . . . . . . 7 (ab) ≤ a
32lecon 154 . . . . . 6 a ≤ (ab)
43leran 153 . . . . 5 (ac ) ≤ ((ab)c )
54lelor 166 . . . 4 (c ∪ (ac )) ≤ (c ∪ ((ab)c ))
61, 5bltr 138 . . 3 (a2 c) ≤ (c ∪ ((ab)c ))
7 df-i2 45 . . . 4 (b2 c) = (c ∪ (bc ))
8 lear 161 . . . . . . 7 (ab) ≤ b
98lecon 154 . . . . . 6 b ≤ (ab)
109leran 153 . . . . 5 (bc ) ≤ ((ab)c )
1110lelor 166 . . . 4 (c ∪ (bc )) ≤ (c ∪ ((ab)c ))
127, 11bltr 138 . . 3 (b2 c) ≤ (c ∪ ((ab)c ))
136, 12lel2or 170 . 2 ((a2 c) ∪ (b2 c)) ≤ (c ∪ ((ab)c ))
14 df-i2 45 . . 3 ((ab) →2 c) = (c ∪ ((ab)c ))
1514ax-r1 35 . 2 (c ∪ ((ab)c )) = ((ab) →2 c)
1613, 15lbtr 139 1 ((a2 c) ∪ (b2 c)) ≤ ((ab) →2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  orbile  843
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