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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 4554. If in the definition of ordinals df-iord 4384, we also required that membership be well-founded on any ordinal (see df-frind 4350), then we could prove ordirr 4559 without ax-setind 4554. (Contributed by NM, 2-Jan-1994.) |
Ref | Expression |
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ordirr |
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Step | Hyp | Ref | Expression |
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1 | elirr 4558 |
. 2
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2 | 1 | a1i 9 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-v 2754 df-dif 3146 df-sn 3613 |
This theorem is referenced by: onirri 4560 nordeq 4561 ordn2lp 4562 orddisj 4563 onprc 4569 nlimsucg 4583 tfr1onlemsucfn 6364 tfr1onlemsucaccv 6365 tfrcllemsucfn 6377 tfrcllemsucaccv 6378 nntr2 6527 unsnfi 6946 nnnninfeq 7155 nninfisol 7160 addnidpig 7364 frecfzennn 10456 hashinfom 10789 hashennn 10791 hashp1i 10821 ennnfonelemg 12453 ctinfom 12478 |
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