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Mirrors > Home > ILE Home > Th. List > elirr | Unicode version |
Description: No class is a member of
itself. Exercise 6 of [TakeutiZaring] p.
22.
The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4452, we could redefine (df-iord 4288) to also require (df-frind 4254) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4457 (which under that definition would presumably not need ax-setind 4452 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4457. To encourage ordirr 4457 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neldifsnd 3654 | . . . . . . . . 9 | |
2 | simp1 981 | . . . . . . . . . . 11 | |
3 | eleq1 2202 | . . . . . . . . . . . . . . . 16 | |
4 | eleq1 2202 | . . . . . . . . . . . . . . . 16 | |
5 | 3, 4 | imbi12d 233 | . . . . . . . . . . . . . . 15 |
6 | 5 | spcgv 2773 | . . . . . . . . . . . . . 14 |
7 | 6 | pm2.43b 52 | . . . . . . . . . . . . 13 |
8 | 7 | 3ad2ant2 1003 | . . . . . . . . . . . 12 |
9 | eleq2 2203 | . . . . . . . . . . . . . 14 | |
10 | 9 | imbi1d 230 | . . . . . . . . . . . . 13 |
11 | 10 | 3ad2ant3 1004 | . . . . . . . . . . . 12 |
12 | 8, 11 | mpbid 146 | . . . . . . . . . . 11 |
13 | 2, 12 | mpd 13 | . . . . . . . . . 10 |
14 | 13 | 3expia 1183 | . . . . . . . . 9 |
15 | 1, 14 | mtod 652 | . . . . . . . 8 |
16 | vex 2689 | . . . . . . . . . 10 | |
17 | eldif 3080 | . . . . . . . . . 10 | |
18 | 16, 17 | mpbiran 924 | . . . . . . . . 9 |
19 | velsn 3544 | . . . . . . . . 9 | |
20 | 18, 19 | xchbinx 671 | . . . . . . . 8 |
21 | 15, 20 | sylibr 133 | . . . . . . 7 |
22 | 21 | ex 114 | . . . . . 6 |
23 | 22 | alrimiv 1846 | . . . . 5 |
24 | df-ral 2421 | . . . . . . . 8 | |
25 | clelsb3 2244 | . . . . . . . . . 10 | |
26 | 25 | imbi2i 225 | . . . . . . . . 9 |
27 | 26 | albii 1446 | . . . . . . . 8 |
28 | 24, 27 | bitri 183 | . . . . . . 7 |
29 | 28 | imbi1i 237 | . . . . . 6 |
30 | 29 | albii 1446 | . . . . 5 |
31 | 23, 30 | sylibr 133 | . . . 4 |
32 | ax-setind 4452 | . . . 4 | |
33 | 31, 32 | syl 14 | . . 3 |
34 | eleq1 2202 | . . . 4 | |
35 | 34 | spcgv 2773 | . . 3 |
36 | 33, 35 | mpd 13 | . 2 |
37 | neldifsnd 3654 | . 2 | |
38 | 36, 37 | pm2.65i 628 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wal 1329 wceq 1331 wcel 1480 wsb 1735 wral 2416 cvv 2686 cdif 3068 csn 3527 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-sn 3533 |
This theorem is referenced by: ordirr 4457 elirrv 4463 sucprcreg 4464 ordsoexmid 4477 onnmin 4483 ssnel 4484 ordtri2or2exmid 4486 reg3exmidlemwe 4493 nntri2 6390 nntri3 6393 nndceq 6395 nndcel 6396 phpelm 6760 fiunsnnn 6775 onunsnss 6805 snon0 6824 |
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