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Mirrors > Home > ILE Home > Th. List > ssdif0im | Unicode version |
Description: Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.) |
Ref | Expression |
---|---|
ssdif0im |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imanim 677 | . . . 4 | |
2 | eldif 3080 | . . . 4 | |
3 | 1, 2 | sylnibr 666 | . . 3 |
4 | 3 | alimi 1431 | . 2 |
5 | dfss2 3086 | . 2 | |
6 | eq0 3381 | . 2 | |
7 | 4, 5, 6 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wal 1329 wceq 1331 wcel 1480 cdif 3068 wss 3071 c0 3363 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 |
This theorem is referenced by: vdif0im 3428 difrab0eqim 3429 difid 3431 difin0 3436 |
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