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

Theorem bj-omtrans 9416
Description: The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4271.

The idea is to use bounded induction with the formula  C_ 
om. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with  C_  a and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

Assertion
Ref Expression
bj-omtrans  om  C_ 
om

Proof of Theorem bj-omtrans
Dummy variables  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-omex 9402 . . 3  om  _V
2 sseq2 2961 . . . . . 6  a  om  C_  a  C_  om
3 sseq2 2961 . . . . . 6  a  om  suc  C_  a  suc  C_ 
om
42, 3imbi12d 223 . . . . 5  a  om  C_  a 
suc  C_  a  C_  om  suc  C_  om
54ralbidv 2320 . . . 4  a  om  om  C_  a 
suc  C_  a  om  C_  om  suc  C_  om
6 sseq2 2961 . . . . 5  a  om  C_  a  C_  om
76imbi2d 219 . . . 4  a  om  om  C_  a  om  C_  om
85, 7imbi12d 223 . . 3  a  om  om  C_  a 
suc  C_  a  om  C_  a 
om  C_  om  suc  C_  om  om  C_  om
9 0ss 3249 . . . 4  (/)  C_  a
10 bdcv 9303 . . . . . 6 BOUNDED  a
1110bdss 9319 . . . . 5 BOUNDED  C_  a
12 nfv 1418 . . . . 5  F/ (/)  C_  a
13 nfv 1418 . . . . 5  F/  C_  a
14 nfv 1418 . . . . 5  F/  suc  C_  a
15 sseq1 2960 . . . . . 6  (/)  C_  a  (/)  C_  a
1615biimprd 147 . . . . 5  (/)  (/)  C_  a  C_  a
17 sseq1 2960 . . . . . 6  C_  a  C_  a
1817biimpd 132 . . . . 5  C_  a  C_  a
19 sseq1 2960 . . . . . 6  suc  C_  a  suc  C_  a
2019biimprd 147 . . . . 5  suc  suc  C_  a  C_  a
21 nfcv 2175 . . . . 5  F/_
22 nfv 1418 . . . . 5  F/  C_  a
23 sseq1 2960 . . . . . 6  C_  a  C_  a
2423biimpd 132 . . . . 5  C_  a  C_  a
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 9408 . . . 4  (/)  C_  a  om  C_  a  suc  C_  a  om  C_  a
269, 25mpan 400 . . 3  om  C_  a  suc  C_  a  om  C_  a
271, 8, 26vtocl 2602 . 2  om  C_  om  suc  C_  om  om  C_  om
28 df-suc 4074 . . . 4  suc  u.  { }
29 simpr 103 . . . . 5  om  C_  om  C_  om
30 simpl 102 . . . . . 6  om  C_  om  om
3130snssd 3500 . . . . 5  om  C_  om  { }  C_  om
3229, 31unssd 3113 . . . 4  om  C_  om  u.  { }  C_  om
3328, 32syl5eqss 2983 . . 3  om  C_  om 
suc  C_  om
3433ex 108 . 2  om  C_  om  suc  C_  om
3527, 34mprg 2372 1  om  C_ 
om
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  wral 2300    u. cun 2909    C_ wss 2911   (/)c0 3218   {csn 3367   suc csuc 4068   omcom 4256
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdor 9271  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386
This theorem is referenced by:  bj-omtrans2  9417  bj-nnord  9418  bj-nn0suc  9424
  Copyright terms: Public domain W3C validator