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Theorem bj-omtrans 10055
Description: The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4328.

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 10041 . . 3  |-  om  e.  _V
2 sseq2 2967 . . . . . 6  |-  ( a  =  om  ->  (
y  C_  a  <->  y  C_  om ) )
3 sseq2 2967 . . . . . 6  |-  ( a  =  om  ->  ( suc  y  C_  a  <->  suc  y  C_  om ) )
42, 3imbi12d 223 . . . . 5  |-  ( a  =  om  ->  (
( y  C_  a  ->  suc  y  C_  a
)  <->  ( y  C_  om 
->  suc  y  C_  om )
) )
54ralbidv 2326 . . . 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 2967 . . . . 5  |-  ( a  =  om  ->  ( A  C_  a  <->  A  C_  om )
)
76imbi2d 219 . . . 4  |-  ( a  =  om  ->  (
( A  e.  om  ->  A  C_  a )  <->  ( A  e.  om  ->  A 
C_  om ) ) )
85, 7imbi12d 223 . . 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 3255 . . . 4  |-  (/)  C_  a
10 bdcv 9942 . . . . . 6  |- BOUNDED  a
1110bdss 9958 . . . . 5  |- BOUNDED  x  C_  a
12 nfv 1421 . . . . 5  |-  F/ x (/)  C_  a
13 nfv 1421 . . . . 5  |-  F/ x  y  C_  a
14 nfv 1421 . . . . 5  |-  F/ x  suc  y  C_  a
15 sseq1 2966 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  a  <->  (/)  C_  a
) )
1615biimprd 147 . . . . 5  |-  ( x  =  (/)  ->  ( (/)  C_  a  ->  x  C_  a
) )
17 sseq1 2966 . . . . . 6  |-  ( x  =  y  ->  (
x  C_  a  <->  y  C_  a ) )
1817biimpd 132 . . . . 5  |-  ( x  =  y  ->  (
x  C_  a  ->  y 
C_  a ) )
19 sseq1 2966 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  C_  a  <->  suc  y  C_  a )
)
2019biimprd 147 . . . . 5  |-  ( x  =  suc  y  -> 
( suc  y  C_  a  ->  x  C_  a
) )
21 nfcv 2178 . . . . 5  |-  F/_ x A
22 nfv 1421 . . . . 5  |-  F/ x  A  C_  a
23 sseq1 2966 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  a  <->  A  C_  a
) )
2423biimpd 132 . . . . 5  |-  ( x  =  A  ->  (
x  C_  a  ->  A 
C_  a ) )
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 10047 . . . 4  |-  ( (
(/)  C_  a  /\  A. y  e.  om  (
y  C_  a  ->  suc  y  C_  a )
)  ->  ( A  e.  om  ->  A  C_  a
) )
269, 25mpan 400 . . 3  |-  ( A. y  e.  om  (
y  C_  a  ->  suc  y  C_  a )  ->  ( A  e.  om  ->  A  C_  a )
)
271, 8, 26vtocl 2608 . 2  |-  ( A. y  e.  om  (
y  C_  om  ->  suc  y  C_  om )  ->  ( A  e.  om  ->  A  C_  om )
)
28 df-suc 4108 . . . 4  |-  suc  y  =  ( y  u. 
{ y } )
29 simpr 103 . . . . 5  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
y  C_  om )
30 simpl 102 . . . . . 6  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
y  e.  om )
3130snssd 3509 . . . . 5  |-  ( ( y  e.  om  /\  y  C_  om )  ->  { y }  C_  om )
3229, 31unssd 3119 . . . 4  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
( y  u.  {
y } )  C_  om )
3328, 32syl5eqss 2989 . . 3  |-  ( ( y  e.  om  /\  y  C_  om )  ->  suc  y  C_  om )
3433ex 108 . 2  |-  ( y  e.  om  ->  (
y  C_  om  ->  suc  y  C_  om )
)
3527, 34mprg 2378 1  |-  ( A  e.  om  ->  A  C_ 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   A.wral 2306    u. cun 2915    C_ wss 2917   (/)c0 3224   {csn 3375   suc csuc 4102   omcom 4313
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9907  ax-bdor 9910  ax-bdal 9912  ax-bdex 9913  ax-bdeq 9914  ax-bdel 9915  ax-bdsb 9916  ax-bdsep 9978  ax-infvn 10040
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9935  df-bj-ind 10025
This theorem is referenced by:  bj-omtrans2  10056  bj-nnord  10057  bj-nn0suc  10063
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