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Mirrors > Home > ILE Home > Th. List > ssundifim | Unicode version |
Description: A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
ssundifim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.6r 912 | . . . 4 | |
2 | elun 3212 | . . . . 5 | |
3 | 2 | imbi2i 225 | . . . 4 |
4 | eldif 3075 | . . . . 5 | |
5 | 4 | imbi1i 237 | . . . 4 |
6 | 1, 3, 5 | 3imtr4i 200 | . . 3 |
7 | 6 | alimi 1431 | . 2 |
8 | dfss2 3081 | . 2 | |
9 | dfss2 3081 | . 2 | |
10 | 7, 8, 9 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wal 1329 wcel 1480 cdif 3063 cun 3064 wss 3066 |
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-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 |
This theorem is referenced by: (None) |
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