Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4422. If in the definition of ordinals df-iord 4258, we also required that membership be well-founded on any ordinal (see df-frind 4224), then we could prove ordirr 4427 without ax-setind 4422. (Contributed by NM, 2-Jan-1994.) |
Ref | Expression |
---|---|
ordirr | ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4426 | . 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 1465 Ord word 4254 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-v 2662 df-dif 3043 df-sn 3503 |
This theorem is referenced by: onirri 4428 nordeq 4429 ordn2lp 4430 orddisj 4431 onprc 4437 nlimsucg 4451 tfr1onlemsucfn 6205 tfr1onlemsucaccv 6206 tfrcllemsucfn 6218 tfrcllemsucaccv 6219 nntr2 6367 unsnfi 6775 addnidpig 7112 frecfzennn 10167 hashinfom 10492 hashennn 10494 hashp1i 10524 ennnfonelemg 11843 ctinfom 11868 nninfalllemn 13129 |
Copyright terms: Public domain | W3C validator |