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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 4456 for natural numbers, which does not require ax-setind 4452. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnelirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3367 | . 2 | |
2 | df-suc 4293 | . . . . . 6 | |
3 | 2 | eleq2i 2206 | . . . . 5 |
4 | elun 3217 | . . . . . 6 | |
5 | bj-nntrans 13149 | . . . . . . . 8 | |
6 | sucssel 4346 | . . . . . . . 8 | |
7 | 5, 6 | syld 45 | . . . . . . 7 |
8 | vex 2689 | . . . . . . . . . 10 | |
9 | 8 | sucid 4339 | . . . . . . . . 9 |
10 | elsni 3545 | . . . . . . . . 9 | |
11 | 9, 10 | eleqtrid 2228 | . . . . . . . 8 |
12 | 11 | a1i 9 | . . . . . . 7 |
13 | 7, 12 | jaod 706 | . . . . . 6 |
14 | 4, 13 | syl5bi 151 | . . . . 5 |
15 | 3, 14 | syl5bi 151 | . . . 4 |
16 | 15 | con3d 620 | . . 3 |
17 | 16 | rgen 2485 | . 2 |
18 | ax-bdel 13019 | . . . 4 BOUNDED | |
19 | 18 | ax-bdn 13015 | . . 3 BOUNDED |
20 | nfv 1508 | . . 3 | |
21 | nfv 1508 | . . 3 | |
22 | nfv 1508 | . . 3 | |
23 | eleq1 2202 | . . . . . 6 | |
24 | eleq2 2203 | . . . . . 6 | |
25 | 23, 24 | bitrd 187 | . . . . 5 |
26 | 25 | notbid 656 | . . . 4 |
27 | 26 | biimprd 157 | . . 3 |
28 | elequ1 1690 | . . . . . 6 | |
29 | elequ2 1691 | . . . . . 6 | |
30 | 28, 29 | bitrd 187 | . . . . 5 |
31 | 30 | notbid 656 | . . . 4 |
32 | 31 | biimpd 143 | . . 3 |
33 | eleq1 2202 | . . . . . 6 | |
34 | eleq2 2203 | . . . . . 6 | |
35 | 33, 34 | bitrd 187 | . . . . 5 |
36 | 35 | notbid 656 | . . . 4 |
37 | 36 | biimprd 157 | . . 3 |
38 | nfcv 2281 | . . 3 | |
39 | nfv 1508 | . . 3 | |
40 | eleq1 2202 | . . . . . 6 | |
41 | eleq2 2203 | . . . . . 6 | |
42 | 40, 41 | bitrd 187 | . . . . 5 |
43 | 42 | notbid 656 | . . . 4 |
44 | 43 | biimpd 143 | . . 3 |
45 | 19, 20, 21, 22, 27, 32, 37, 38, 39, 44 | bj-bdfindisg 13146 | . 2 |
46 | 1, 17, 45 | mp2an 422 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 697 wceq 1331 wcel 1480 wral 2416 cun 3069 wss 3071 c0 3363 csn 3527 csuc 4287 com 4504 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 ax-pr 4131 ax-un 4355 ax-bd0 13011 ax-bdor 13014 ax-bdn 13015 ax-bdal 13016 ax-bdex 13017 ax-bdeq 13018 ax-bdel 13019 ax-bdsb 13020 ax-bdsep 13082 ax-infvn 13139 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 df-bdc 13039 df-bj-ind 13125 |
This theorem is referenced by: bj-nnen2lp 13152 |
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