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Description: The set is transitive. A
natural number is included in
.
Constructive proof of elnn 4519.
The idea is to use bounded induction with the formula . This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omtrans |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omex 13140 | . . 3 | |
2 | sseq2 3121 | . . . . . 6 | |
3 | sseq2 3121 | . . . . . 6 | |
4 | 2, 3 | imbi12d 233 | . . . . 5 |
5 | 4 | ralbidv 2437 | . . . 4 |
6 | sseq2 3121 | . . . . 5 | |
7 | 6 | imbi2d 229 | . . . 4 |
8 | 5, 7 | imbi12d 233 | . . 3 |
9 | 0ss 3401 | . . . 4 | |
10 | bdcv 13046 | . . . . . 6 BOUNDED | |
11 | 10 | bdss 13062 | . . . . 5 BOUNDED |
12 | nfv 1508 | . . . . 5 | |
13 | nfv 1508 | . . . . 5 | |
14 | nfv 1508 | . . . . 5 | |
15 | sseq1 3120 | . . . . . 6 | |
16 | 15 | biimprd 157 | . . . . 5 |
17 | sseq1 3120 | . . . . . 6 | |
18 | 17 | biimpd 143 | . . . . 5 |
19 | sseq1 3120 | . . . . . 6 | |
20 | 19 | biimprd 157 | . . . . 5 |
21 | nfcv 2281 | . . . . 5 | |
22 | nfv 1508 | . . . . 5 | |
23 | sseq1 3120 | . . . . . 6 | |
24 | 23 | biimpd 143 | . . . . 5 |
25 | 11, 12, 13, 14, 16, 18, 20, 21, 22, 24 | bj-bdfindisg 13146 | . . . 4 |
26 | 9, 25 | mpan 420 | . . 3 |
27 | 1, 8, 26 | vtocl 2740 | . 2 |
28 | df-suc 4293 | . . . 4 | |
29 | simpr 109 | . . . . 5 | |
30 | simpl 108 | . . . . . 6 | |
31 | 30 | snssd 3665 | . . . . 5 |
32 | 29, 31 | unssd 3252 | . . . 4 |
33 | 28, 32 | eqsstrid 3143 | . . 3 |
34 | 33 | ex 114 | . 2 |
35 | 27, 34 | mprg 2489 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 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-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-omtrans2 13155 bj-nnord 13156 bj-nn0suc 13162 |
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