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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 3636 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 | elnn 4489 | . . . . . 6 | |
2 | 1 | expcom 115 | . . . . 5 |
3 | elnn 4489 | . . . . . 6 | |
4 | 3 | expcom 115 | . . . . 5 |
5 | 2, 4 | anim12d 333 | . . . 4 |
6 | nndceq 6363 | . . . 4 DECID | |
7 | 5, 6 | syl6 33 | . . 3 DECID |
8 | 7 | ralrimivv 2490 | . 2 DECID |
9 | dcdifsnid 6368 | . 2 DECID | |
10 | 8, 9 | sylan 281 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 804 wceq 1316 wcel 1465 wral 2393 cdif 3038 cun 3039 csn 3497 com 4474 |
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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 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-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 |
This theorem is referenced by: phplem2 6715 |
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