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Mirrors > Home > ILE Home > Th. List > nnsseleq | Unicode version |
Description: For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.) |
Ref | Expression |
---|---|
nnsseleq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri1 6074 | . . 3 | |
2 | nntri3or 6072 | . . . . . 6 | |
3 | df-3or 886 | . . . . . 6 | |
4 | 2, 3 | sylib 127 | . . . . 5 |
5 | 4 | orcomd 648 | . . . 4 |
6 | 5 | ord 643 | . . 3 |
7 | 1, 6 | sylbid 139 | . 2 |
8 | nnord 4334 | . . . . 5 | |
9 | 8 | adantl 262 | . . . 4 |
10 | ordelss 4116 | . . . . 5 | |
11 | 10 | ex 108 | . . . 4 |
12 | 9, 11 | syl 14 | . . 3 |
13 | eqimss 2997 | . . . 4 | |
14 | 13 | a1i 9 | . . 3 |
15 | 12, 14 | jaod 637 | . 2 |
16 | 7, 15 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3o 884 wceq 1243 wcel 1393 wss 2917 word 4099 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-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: (None) |
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