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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nntrans | Unicode version |
Description: A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nntrans |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 3322 | . . 3 | |
2 | df-suc 4108 | . . . . . . 7 | |
3 | 2 | eleq2i 2104 | . . . . . 6 |
4 | elun 3084 | . . . . . . 7 | |
5 | sssucid 4152 | . . . . . . . . . 10 | |
6 | sstr2 2952 | . . . . . . . . . 10 | |
7 | 5, 6 | mpi 15 | . . . . . . . . 9 |
8 | 7 | imim2i 12 | . . . . . . . 8 |
9 | elsni 3393 | . . . . . . . . . 10 | |
10 | 9, 5 | syl6eqss 2995 | . . . . . . . . 9 |
11 | 10 | a1i 9 | . . . . . . . 8 |
12 | 8, 11 | jaod 637 | . . . . . . 7 |
13 | 4, 12 | syl5bi 141 | . . . . . 6 |
14 | 3, 13 | syl5bi 141 | . . . . 5 |
15 | 14 | ralimi2 2381 | . . . 4 |
16 | 15 | rgenw 2376 | . . 3 |
17 | bdcv 9968 | . . . . . 6 BOUNDED | |
18 | 17 | bdss 9984 | . . . . 5 BOUNDED |
19 | 18 | ax-bdal 9938 | . . . 4 BOUNDED |
20 | nfv 1421 | . . . 4 | |
21 | nfv 1421 | . . . 4 | |
22 | nfv 1421 | . . . 4 | |
23 | sseq2 2967 | . . . . . 6 | |
24 | 23 | raleqbi1dv 2513 | . . . . 5 |
25 | 24 | biimprd 147 | . . . 4 |
26 | sseq2 2967 | . . . . . 6 | |
27 | 26 | raleqbi1dv 2513 | . . . . 5 |
28 | 27 | biimpd 132 | . . . 4 |
29 | sseq2 2967 | . . . . . 6 | |
30 | 29 | raleqbi1dv 2513 | . . . . 5 |
31 | 30 | biimprd 147 | . . . 4 |
32 | nfcv 2178 | . . . 4 | |
33 | nfv 1421 | . . . 4 | |
34 | sseq2 2967 | . . . . . 6 | |
35 | 34 | raleqbi1dv 2513 | . . . . 5 |
36 | 35 | biimpd 132 | . . . 4 |
37 | 19, 20, 21, 22, 25, 28, 31, 32, 33, 36 | bj-bdfindisg 10073 | . . 3 |
38 | 1, 16, 37 | mp2an 402 | . 2 |
39 | nfv 1421 | . . 3 | |
40 | sseq1 2966 | . . 3 | |
41 | 39, 40 | rspc 2650 | . 2 |
42 | 38, 41 | syl5com 26 | 1 |
Colors of variables: wff set class |
Syntax hints: 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-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-nntrans2 10077 bj-nnelirr 10078 bj-nnen2lp 10079 |
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