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Theorem bj-omtrans 10909
Description: The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4354.

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

Assertion
Ref Expression
bj-omtrans  |-  ( A  e.  om  ->  A  C_ 
om )

Proof of Theorem bj-omtrans
Dummy variables  x  a  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-omex 10895 . . 3  |-  om  e.  _V
2 sseq2 3022 . . . . . 6  |-  ( a  =  om  ->  (
y  C_  a  <->  y  C_  om ) )
3 sseq2 3022 . . . . . 6  |-  ( a  =  om  ->  ( suc  y  C_  a  <->  suc  y  C_  om ) )
42, 3imbi12d 232 . . . . 5  |-  ( a  =  om  ->  (
( y  C_  a  ->  suc  y  C_  a
)  <->  ( y  C_  om 
->  suc  y  C_  om )
) )
54ralbidv 2369 . . . 4  |-  ( a  =  om  ->  ( A. y  e.  om  ( y  C_  a  ->  suc  y  C_  a
)  <->  A. y  e.  om  ( y  C_  om  ->  suc  y  C_  om )
) )
6 sseq2 3022 . . . . 5  |-  ( a  =  om  ->  ( A  C_  a  <->  A  C_  om )
)
76imbi2d 228 . . . 4  |-  ( a  =  om  ->  (
( A  e.  om  ->  A  C_  a )  <->  ( A  e.  om  ->  A 
C_  om ) ) )
85, 7imbi12d 232 . . 3  |-  ( a  =  om  ->  (
( A. y  e. 
om  ( y  C_  a  ->  suc  y  C_  a )  ->  ( A  e.  om  ->  A 
C_  a ) )  <-> 
( A. y  e. 
om  ( y  C_  om 
->  suc  y  C_  om )  ->  ( A  e.  om  ->  A  C_  om )
) ) )
9 0ss 3289 . . . 4  |-  (/)  C_  a
10 bdcv 10797 . . . . . 6  |- BOUNDED  a
1110bdss 10813 . . . . 5  |- BOUNDED  x  C_  a
12 nfv 1462 . . . . 5  |-  F/ x (/)  C_  a
13 nfv 1462 . . . . 5  |-  F/ x  y  C_  a
14 nfv 1462 . . . . 5  |-  F/ x  suc  y  C_  a
15 sseq1 3021 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  a  <->  (/)  C_  a
) )
1615biimprd 156 . . . . 5  |-  ( x  =  (/)  ->  ( (/)  C_  a  ->  x  C_  a
) )
17 sseq1 3021 . . . . . 6  |-  ( x  =  y  ->  (
x  C_  a  <->  y  C_  a ) )
1817biimpd 142 . . . . 5  |-  ( x  =  y  ->  (
x  C_  a  ->  y 
C_  a ) )
19 sseq1 3021 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  C_  a  <->  suc  y  C_  a )
)
2019biimprd 156 . . . . 5  |-  ( x  =  suc  y  -> 
( suc  y  C_  a  ->  x  C_  a
) )
21 nfcv 2220 . . . . 5  |-  F/_ x A
22 nfv 1462 . . . . 5  |-  F/ x  A  C_  a
23 sseq1 3021 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  a  <->  A  C_  a
) )
2423biimpd 142 . . . . 5  |-  ( x  =  A  ->  (
x  C_  a  ->  A 
C_  a ) )
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 10901 . . . 4  |-  ( (
(/)  C_  a  /\  A. y  e.  om  (
y  C_  a  ->  suc  y  C_  a )
)  ->  ( A  e.  om  ->  A  C_  a
) )
269, 25mpan 415 . . 3  |-  ( A. y  e.  om  (
y  C_  a  ->  suc  y  C_  a )  ->  ( A  e.  om  ->  A  C_  a )
)
271, 8, 26vtocl 2654 . 2  |-  ( A. y  e.  om  (
y  C_  om  ->  suc  y  C_  om )  ->  ( A  e.  om  ->  A  C_  om )
)
28 df-suc 4134 . . . 4  |-  suc  y  =  ( y  u. 
{ y } )
29 simpr 108 . . . . 5  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
y  C_  om )
30 simpl 107 . . . . . 6  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
y  e.  om )
3130snssd 3538 . . . . 5  |-  ( ( y  e.  om  /\  y  C_  om )  ->  { y }  C_  om )
3229, 31unssd 3149 . . . 4  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
( y  u.  {
y } )  C_  om )
3328, 32syl5eqss 3044 . . 3  |-  ( ( y  e.  om  /\  y  C_  om )  ->  suc  y  C_  om )
3433ex 113 . 2  |-  ( y  e.  om  ->  (
y  C_  om  ->  suc  y  C_  om )
)
3527, 34mprg 2421 1  |-  ( A  e.  om  ->  A  C_ 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349    u. cun 2972    C_ wss 2974   (/)c0 3258   {csn 3406   suc csuc 4128   omcom 4339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10762  ax-bdor 10765  ax-bdal 10767  ax-bdex 10768  ax-bdeq 10769  ax-bdel 10770  ax-bdsb 10771  ax-bdsep 10833  ax-infvn 10894
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-bdc 10790  df-bj-ind 10880
This theorem is referenced by:  bj-omtrans2  10910  bj-nnord  10911  bj-nn0suc  10917
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