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Mirrors > Home > ILE Home > Th. List > nndifsnid | Unicode version |
Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3510 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.) |
Ref | Expression |
---|---|
nndifsnid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsnss 3510 | . . 3 | |
2 | 1 | adantl 262 | . 2 |
3 | simpr 103 | . . . . . . 7 | |
4 | velsn 3392 | . . . . . . 7 | |
5 | 3, 4 | sylibr 137 | . . . . . 6 |
6 | elun2 3111 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | simplr 482 | . . . . . . 7 | |
9 | simpr 103 | . . . . . . . 8 | |
10 | 9, 4 | sylnibr 602 | . . . . . . 7 |
11 | 8, 10 | eldifd 2928 | . . . . . 6 |
12 | elun1 3110 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | simpr 103 | . . . . . . . 8 | |
15 | simpll 481 | . . . . . . . 8 | |
16 | elnn 4328 | . . . . . . . 8 | |
17 | 14, 15, 16 | syl2anc 391 | . . . . . . 7 |
18 | simplr 482 | . . . . . . . 8 | |
19 | elnn 4328 | . . . . . . . 8 | |
20 | 18, 15, 19 | syl2anc 391 | . . . . . . 7 |
21 | nndceq 6077 | . . . . . . 7 DECID | |
22 | 17, 20, 21 | syl2anc 391 | . . . . . 6 DECID |
23 | df-dc 743 | . . . . . 6 DECID | |
24 | 22, 23 | sylib 127 | . . . . 5 |
25 | 7, 13, 24 | mpjaodan 711 | . . . 4 |
26 | 25 | ex 108 | . . 3 |
27 | 26 | ssrdv 2951 | . 2 |
28 | 2, 27 | eqssd 2962 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 DECID wdc 742 wceq 1243 wcel 1393 cdif 2914 cun 2915 wss 2917 csn 3375 com 4313 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 |
This theorem is referenced by: phplem2 6316 |
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