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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 4266 for natural numbers, which does not require ax-setind 4262. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnelirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3228 | . 2 | |
2 | df-suc 4108 | . . . . . 6 | |
3 | 2 | eleq2i 2104 | . . . . 5 |
4 | elun 3084 | . . . . . 6 | |
5 | bj-nntrans 10076 | . . . . . . . 8 | |
6 | sucssel 4161 | . . . . . . . 8 | |
7 | 5, 6 | syld 40 | . . . . . . 7 |
8 | vex 2560 | . . . . . . . . . 10 | |
9 | 8 | sucid 4154 | . . . . . . . . 9 |
10 | elsni 3393 | . . . . . . . . 9 | |
11 | 9, 10 | syl5eleq 2126 | . . . . . . . 8 |
12 | 11 | a1i 9 | . . . . . . 7 |
13 | 7, 12 | jaod 637 | . . . . . 6 |
14 | 4, 13 | syl5bi 141 | . . . . 5 |
15 | 3, 14 | syl5bi 141 | . . . 4 |
16 | 15 | con3d 561 | . . 3 |
17 | 16 | rgen 2374 | . 2 |
18 | ax-bdel 9941 | . . . 4 BOUNDED | |
19 | 18 | ax-bdn 9937 | . . 3 BOUNDED |
20 | nfv 1421 | . . 3 | |
21 | nfv 1421 | . . 3 | |
22 | nfv 1421 | . . 3 | |
23 | eleq1 2100 | . . . . . 6 | |
24 | eleq2 2101 | . . . . . 6 | |
25 | 23, 24 | bitrd 177 | . . . . 5 |
26 | 25 | notbid 592 | . . . 4 |
27 | 26 | biimprd 147 | . . 3 |
28 | elequ1 1600 | . . . . . 6 | |
29 | elequ2 1601 | . . . . . 6 | |
30 | 28, 29 | bitrd 177 | . . . . 5 |
31 | 30 | notbid 592 | . . . 4 |
32 | 31 | biimpd 132 | . . 3 |
33 | eleq1 2100 | . . . . . 6 | |
34 | eleq2 2101 | . . . . . 6 | |
35 | 33, 34 | bitrd 177 | . . . . 5 |
36 | 35 | notbid 592 | . . . 4 |
37 | 36 | biimprd 147 | . . 3 |
38 | nfcv 2178 | . . 3 | |
39 | nfv 1421 | . . 3 | |
40 | eleq1 2100 | . . . . . 6 | |
41 | eleq2 2101 | . . . . . 6 | |
42 | 40, 41 | bitrd 177 | . . . . 5 |
43 | 42 | notbid 592 | . . . 4 |
44 | 43 | biimpd 132 | . . 3 |
45 | 19, 20, 21, 22, 27, 32, 37, 38, 39, 44 | bj-bdfindisg 10073 | . 2 |
46 | 1, 17, 45 | mp2an 402 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 629 wceq 1243 wcel 1393 wral 2306 cun 2915 wss 2917 c0 3224 csn 3375 csuc 4102 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-nul 3883 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdor 9936 ax-bdn 9937 ax-bdal 9938 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 ax-infvn 10066 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bj-nnen2lp 10079 |
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