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Mirrors > Home > ILE Home > Th. List > ordirr | GIF 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 4535. If in the definition of ordinals df-iord 4365, we also required that membership be well-founded on any ordinal (see df-frind 4331), then we could prove ordirr 4540 without ax-setind 4535. (Contributed by NM, 2-Jan-1994.) |
Ref | Expression |
---|---|
ordirr | ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4539 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | 1 | a1i 9 | 1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2148 Ord word 4361 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-setind 4535 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-v 2739 df-dif 3131 df-sn 3598 |
This theorem is referenced by: onirri 4541 nordeq 4542 ordn2lp 4543 orddisj 4544 onprc 4550 nlimsucg 4564 tfr1onlemsucfn 6338 tfr1onlemsucaccv 6339 tfrcllemsucfn 6351 tfrcllemsucaccv 6352 nntr2 6501 unsnfi 6915 nnnninfeq 7123 nninfisol 7128 addnidpig 7332 frecfzennn 10421 hashinfom 10751 hashennn 10753 hashp1i 10783 ennnfonelemg 12396 ctinfom 12421 |
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