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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nnelirr | Unicode version |
Description: A natural number does not belong to itself. Version of elirr 4224 for natural numbers, which does not require ax-setind 4220. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnelirr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3222 |
. 2
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2 | df-suc 4074 |
. . . . . 6
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3 | 2 | eleq2i 2101 |
. . . . 5
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4 | elun 3078 |
. . . . . 6
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5 | bj-nntrans 9411 |
. . . . . . . 8
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6 | sucssel 4127 |
. . . . . . . 8
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7 | 5, 6 | syld 40 |
. . . . . . 7
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8 | vex 2554 |
. . . . . . . . . 10
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9 | 8 | sucid 4120 |
. . . . . . . . 9
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10 | elsni 3391 |
. . . . . . . . 9
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11 | 9, 10 | syl5eleq 2123 |
. . . . . . . 8
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12 | 11 | a1i 9 |
. . . . . . 7
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13 | 7, 12 | jaod 636 |
. . . . . 6
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14 | 4, 13 | syl5bi 141 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 3, 14 | syl5bi 141 |
. . . 4
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16 | 15 | con3d 560 |
. . 3
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17 | 16 | rgen 2368 |
. 2
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18 | ax-bdel 9276 |
. . . 4
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19 | 18 | ax-bdn 9272 |
. . 3
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20 | nfv 1418 |
. . 3
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21 | nfv 1418 |
. . 3
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22 | nfv 1418 |
. . 3
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23 | eleq1 2097 |
. . . . . 6
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24 | eleq2 2098 |
. . . . . 6
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25 | 23, 24 | bitrd 177 |
. . . . 5
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26 | 25 | notbid 591 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | biimprd 147 |
. . 3
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28 | elequ1 1597 |
. . . . . 6
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29 | elequ2 1598 |
. . . . . 6
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30 | 28, 29 | bitrd 177 |
. . . . 5
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31 | 30 | notbid 591 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31 | biimpd 132 |
. . 3
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33 | eleq1 2097 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | eleq2 2098 |
. . . . . 6
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35 | 33, 34 | bitrd 177 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 35 | notbid 591 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | biimprd 147 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | nfcv 2175 |
. . 3
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39 | nfv 1418 |
. . 3
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40 | eleq1 2097 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | eleq2 2098 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
42 | 40, 41 | bitrd 177 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 42 | notbid 591 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 43 | biimpd 132 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 19, 20, 21, 22, 27, 32, 37, 38, 39, 44 | bj-bdfindisg 9408 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 1, 17, 45 | mp2an 402 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdor 9271 ax-bdn 9272 ax-bdal 9273 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 ax-infvn 9401 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: bj-nnen2lp 9414 |
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