Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-omtrans Unicode version

Theorem bj-omtrans 13477
 Description: The set is transitive. A natural number is included in . Constructive proof of elnn 4559. 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.)
Assertion
Ref Expression
bj-omtrans

Proof of Theorem bj-omtrans
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-omex 13463 . . 3
2 sseq2 3148 . . . . . 6
3 sseq2 3148 . . . . . 6
42, 3imbi12d 233 . . . . 5
54ralbidv 2454 . . . 4
6 sseq2 3148 . . . . 5
76imbi2d 229 . . . 4
85, 7imbi12d 233 . . 3
9 0ss 3428 . . . 4
10 bdcv 13369 . . . . . 6 BOUNDED
1110bdss 13385 . . . . 5 BOUNDED
12 nfv 1505 . . . . 5
13 nfv 1505 . . . . 5
14 nfv 1505 . . . . 5
15 sseq1 3147 . . . . . 6
1615biimprd 157 . . . . 5
17 sseq1 3147 . . . . . 6
1817biimpd 143 . . . . 5
19 sseq1 3147 . . . . . 6
2019biimprd 157 . . . . 5
21 nfcv 2296 . . . . 5
22 nfv 1505 . . . . 5
23 sseq1 3147 . . . . . 6
2423biimpd 143 . . . . 5
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 13469 . . . 4
269, 25mpan 421 . . 3
271, 8, 26vtocl 2763 . 2
28 df-suc 4326 . . . 4
29 simpr 109 . . . . 5
30 simpl 108 . . . . . 6
3130snssd 3697 . . . . 5
3229, 31unssd 3279 . . . 4
3328, 32eqsstrid 3170 . . 3
3433ex 114 . 2
3527, 34mprg 2511 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1332   wcel 2125  wral 2432   cun 3096   wss 3098  c0 3390  csn 3556   csuc 4320  com 4543 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-nul 4086  ax-pr 4164  ax-un 4388  ax-bd0 13334  ax-bdor 13337  ax-bdal 13339  ax-bdex 13340  ax-bdeq 13341  ax-bdel 13342  ax-bdsb 13343  ax-bdsep 13405  ax-infvn 13462 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-sn 3562  df-pr 3563  df-uni 3769  df-int 3804  df-suc 4326  df-iom 4544  df-bdc 13362  df-bj-ind 13448 This theorem is referenced by:  bj-omtrans2  13478  bj-nnord  13479  bj-nn0suc  13485
 Copyright terms: Public domain W3C validator