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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 4447, we could redefine (df-iord 4283) to also require (df-frind 4249) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4452 (which under that definition would presumably not need ax-setind 4447 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 4452. To encourage ordirr 4452 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 3649 | . . . . . . . . 9 | |
2 | simp1 981 | . . . . . . . . . . 11 | |
3 | eleq1 2200 | . . . . . . . . . . . . . . . 16 | |
4 | eleq1 2200 | . . . . . . . . . . . . . . . 16 | |
5 | 3, 4 | imbi12d 233 | . . . . . . . . . . . . . . 15 |
6 | 5 | spcgv 2768 | . . . . . . . . . . . . . 14 |
7 | 6 | pm2.43b 52 | . . . . . . . . . . . . 13 |
8 | 7 | 3ad2ant2 1003 | . . . . . . . . . . . 12 |
9 | eleq2 2201 | . . . . . . . . . . . . . 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 2684 | . . . . . . . . . 10 | |
17 | eldif 3075 | . . . . . . . . . 10 | |
18 | 16, 17 | mpbiran 924 | . . . . . . . . 9 |
19 | velsn 3539 | . . . . . . . . 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 2419 | . . . . . . . 8 | |
25 | clelsb3 2242 | . . . . . . . . . 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 4447 | . . . 4 | |
33 | 31, 32 | syl 14 | . . 3 |
34 | eleq1 2200 | . . . 4 | |
35 | 34 | spcgv 2768 | . . 3 |
36 | 33, 35 | mpd 13 | . 2 |
37 | neldifsnd 3649 | . 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 2414 cvv 2681 cdif 3063 csn 3522 |
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 2119 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-v 2683 df-dif 3068 df-sn 3528 |
This theorem is referenced by: ordirr 4452 elirrv 4458 sucprcreg 4459 ordsoexmid 4472 onnmin 4478 ssnel 4479 ordtri2or2exmid 4481 reg3exmidlemwe 4488 nntri2 6383 nntri3 6386 nndceq 6388 nndcel 6389 phpelm 6753 fiunsnnn 6768 onunsnss 6798 snon0 6817 |
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