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Mirrors > Home > ILE Home > Th. List > undif4 | Unicode version |
Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
undif4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.621 736 | . . . . . . 7 | |
2 | olc 700 | . . . . . . 7 | |
3 | 1, 2 | impbid1 141 | . . . . . 6 |
4 | 3 | anbi2d 459 | . . . . 5 |
5 | eldif 3075 | . . . . . . 7 | |
6 | 5 | orbi2i 751 | . . . . . 6 |
7 | ordi 805 | . . . . . 6 | |
8 | 6, 7 | bitri 183 | . . . . 5 |
9 | elun 3212 | . . . . . 6 | |
10 | 9 | anbi1i 453 | . . . . 5 |
11 | 4, 8, 10 | 3bitr4g 222 | . . . 4 |
12 | elun 3212 | . . . 4 | |
13 | eldif 3075 | . . . 4 | |
14 | 11, 12, 13 | 3bitr4g 222 | . . 3 |
15 | 14 | alimi 1431 | . 2 |
16 | disj1 3408 | . 2 | |
17 | dfcleq 2131 | . 2 | |
18 | 15, 16, 17 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wal 1329 wceq 1331 wcel 1480 cdif 3063 cun 3064 cin 3065 c0 3358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-nul 3359 |
This theorem is referenced by: phplem1 6739 |
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