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Mirrors > Home > ILE Home > Th. List > ssdifeq0 | Unicode version |
Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
Ref | Expression |
---|---|
ssdifeq0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 3316 | . . 3 | |
2 | ssdifin0 3475 | . . 3 | |
3 | 1, 2 | eqtr3id 2204 | . 2 |
4 | 0ss 3432 | . . 3 | |
5 | id 19 | . . . 4 | |
6 | difeq2 3219 | . . . 4 | |
7 | 5, 6 | sseq12d 3159 | . . 3 |
8 | 4, 7 | mpbiri 167 | . 2 |
9 | 3, 8 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1335 cdif 3099 cin 3101 wss 3102 c0 3394 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rab 2444 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 |
This theorem is referenced by: (None) |
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