Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 3505 | . . 3 | |
2 | df-suc 4343 | . . . . . . 7 | |
3 | 2 | eleq2i 2231 | . . . . . 6 |
4 | elun 3258 | . . . . . . 7 | |
5 | sssucid 4387 | . . . . . . . . . 10 | |
6 | sstr2 3144 | . . . . . . . . . 10 | |
7 | 5, 6 | mpi 15 | . . . . . . . . 9 |
8 | 7 | imim2i 12 | . . . . . . . 8 |
9 | elsni 3588 | . . . . . . . . . 10 | |
10 | 9, 5 | eqsstrdi 3189 | . . . . . . . . 9 |
11 | 10 | a1i 9 | . . . . . . . 8 |
12 | 8, 11 | jaod 707 | . . . . . . 7 |
13 | 4, 12 | syl5bi 151 | . . . . . 6 |
14 | 3, 13 | syl5bi 151 | . . . . 5 |
15 | 14 | ralimi2 2524 | . . . 4 |
16 | 15 | rgenw 2519 | . . 3 |
17 | bdcv 13571 | . . . . . 6 BOUNDED | |
18 | 17 | bdss 13587 | . . . . 5 BOUNDED |
19 | 18 | ax-bdal 13541 | . . . 4 BOUNDED |
20 | nfv 1515 | . . . 4 | |
21 | nfv 1515 | . . . 4 | |
22 | nfv 1515 | . . . 4 | |
23 | sseq2 3161 | . . . . . 6 | |
24 | 23 | raleqbi1dv 2667 | . . . . 5 |
25 | 24 | biimprd 157 | . . . 4 |
26 | sseq2 3161 | . . . . . 6 | |
27 | 26 | raleqbi1dv 2667 | . . . . 5 |
28 | 27 | biimpd 143 | . . . 4 |
29 | sseq2 3161 | . . . . . 6 | |
30 | 29 | raleqbi1dv 2667 | . . . . 5 |
31 | 30 | biimprd 157 | . . . 4 |
32 | nfcv 2306 | . . . 4 | |
33 | nfv 1515 | . . . 4 | |
34 | sseq2 3161 | . . . . . 6 | |
35 | 34 | raleqbi1dv 2667 | . . . . 5 |
36 | 35 | biimpd 143 | . . . 4 |
37 | 19, 20, 21, 22, 25, 28, 31, 32, 33, 36 | bj-bdfindisg 13671 | . . 3 |
38 | 1, 16, 37 | mp2an 423 | . 2 |
39 | nfv 1515 | . . 3 | |
40 | sseq1 3160 | . . 3 | |
41 | 39, 40 | rspc 2819 | . 2 |
42 | 38, 41 | syl5com 29 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 698 wceq 1342 wcel 2135 wral 2442 cun 3109 wss 3111 c0 3404 csn 3570 csuc 4337 com 4561 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4102 ax-pr 4181 ax-un 4405 ax-bd0 13536 ax-bdor 13539 ax-bdal 13541 ax-bdex 13542 ax-bdeq 13543 ax-bdel 13544 ax-bdsb 13545 ax-bdsep 13607 ax-infvn 13664 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 df-bdc 13564 df-bj-ind 13650 |
This theorem is referenced by: bj-nntrans2 13675 bj-nnelirr 13676 bj-nnen2lp 13677 |
Copyright terms: Public domain | W3C validator |