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Mirrors > Home > QLE Home > Th. List > i3orlem3 | GIF version |
Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.) |
Ref | Expression |
---|---|
i3orlem3 | c ≤ ((a ∪ c) →3 (b ∪ c)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-a2 31 | . . . . . 6 ((a ∪ c)⊥ ∪ c) = (c ∪ (a ∪ c)⊥ ) | |
2 | 1 | lan 77 | . . . . 5 (c ∩ ((a ∪ c)⊥ ∪ c)) = (c ∩ (c ∪ (a ∪ c)⊥ )) |
3 | anabs 121 | . . . . 5 (c ∩ (c ∪ (a ∪ c)⊥ )) = c | |
4 | 2, 3 | ax-r2 36 | . . . 4 (c ∩ ((a ∪ c)⊥ ∪ c)) = c |
5 | 4 | ax-r1 35 | . . 3 c = (c ∩ ((a ∪ c)⊥ ∪ c)) |
6 | leor 159 | . . . 4 c ≤ (a ∪ c) | |
7 | leor 159 | . . . . 5 c ≤ (b ∪ c) | |
8 | 7 | lelor 166 | . . . 4 ((a ∪ c)⊥ ∪ c) ≤ ((a ∪ c)⊥ ∪ (b ∪ c)) |
9 | 6, 8 | le2an 169 | . . 3 (c ∩ ((a ∪ c)⊥ ∪ c)) ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) |
10 | 5, 9 | bltr 138 | . 2 c ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) |
11 | i3orlem1 552 | . 2 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c)) | |
12 | 10, 11 | letr 137 | 1 c ≤ ((a ∪ c) →3 (b ∪ c)) |
Colors of variables: term |
Syntax hints: ≤ wle 2 ⊥ wn 4 ∪ wo 6 ∩ wa 7 →3 wi3 14 |
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-i3 46 df-le1 130 df-le2 131 |
This theorem is referenced by: (None) |
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