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Mirrors > Home > ILE Home > Th. List > ordirr | Unicode version |
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4280. If in the definition of ordinals df-iord 4121, we also required that membership be well-founded on any ordinal (see df-frind 4087), then we could prove ordirr 4285 without ax-setind 4280. (Contributed by NM, 2-Jan-1994.) |
Ref | Expression |
---|---|
ordirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4284 | . 2 | |
2 | 1 | a1i 9 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wcel 1433 word 4117 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-sn 3404 |
This theorem is referenced by: onirri 4286 nordeq 4287 ordn2lp 4288 orddisj 4289 onprc 4295 nlimsucg 4309 unsnfi 6384 addnidpig 6526 frecfzennn 9419 |
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