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Theorem bj-omtrans 12988
Description: The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4487.

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 12974 . . 3  |-  om  e.  _V
2 sseq2 3089 . . . . . 6  |-  ( a  =  om  ->  (
y  C_  a  <->  y  C_  om ) )
3 sseq2 3089 . . . . . 6  |-  ( a  =  om  ->  ( suc  y  C_  a  <->  suc  y  C_  om ) )
42, 3imbi12d 233 . . . . 5  |-  ( a  =  om  ->  (
( y  C_  a  ->  suc  y  C_  a
)  <->  ( y  C_  om 
->  suc  y  C_  om )
) )
54ralbidv 2412 . . . 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 3089 . . . . 5  |-  ( a  =  om  ->  ( A  C_  a  <->  A  C_  om )
)
76imbi2d 229 . . . 4  |-  ( a  =  om  ->  (
( A  e.  om  ->  A  C_  a )  <->  ( A  e.  om  ->  A 
C_  om ) ) )
85, 7imbi12d 233 . . 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 3369 . . . 4  |-  (/)  C_  a
10 bdcv 12880 . . . . . 6  |- BOUNDED  a
1110bdss 12896 . . . . 5  |- BOUNDED  x  C_  a
12 nfv 1491 . . . . 5  |-  F/ x (/)  C_  a
13 nfv 1491 . . . . 5  |-  F/ x  y  C_  a
14 nfv 1491 . . . . 5  |-  F/ x  suc  y  C_  a
15 sseq1 3088 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  a  <->  (/)  C_  a
) )
1615biimprd 157 . . . . 5  |-  ( x  =  (/)  ->  ( (/)  C_  a  ->  x  C_  a
) )
17 sseq1 3088 . . . . . 6  |-  ( x  =  y  ->  (
x  C_  a  <->  y  C_  a ) )
1817biimpd 143 . . . . 5  |-  ( x  =  y  ->  (
x  C_  a  ->  y 
C_  a ) )
19 sseq1 3088 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  C_  a  <->  suc  y  C_  a )
)
2019biimprd 157 . . . . 5  |-  ( x  =  suc  y  -> 
( suc  y  C_  a  ->  x  C_  a
) )
21 nfcv 2256 . . . . 5  |-  F/_ x A
22 nfv 1491 . . . . 5  |-  F/ x  A  C_  a
23 sseq1 3088 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  a  <->  A  C_  a
) )
2423biimpd 143 . . . . 5  |-  ( x  =  A  ->  (
x  C_  a  ->  A 
C_  a ) )
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 12980 . . . 4  |-  ( (
(/)  C_  a  /\  A. y  e.  om  (
y  C_  a  ->  suc  y  C_  a )
)  ->  ( A  e.  om  ->  A  C_  a
) )
269, 25mpan 418 . . 3  |-  ( A. y  e.  om  (
y  C_  a  ->  suc  y  C_  a )  ->  ( A  e.  om  ->  A  C_  a )
)
271, 8, 26vtocl 2712 . 2  |-  ( A. y  e.  om  (
y  C_  om  ->  suc  y  C_  om )  ->  ( A  e.  om  ->  A  C_  om )
)
28 df-suc 4261 . . . 4  |-  suc  y  =  ( y  u. 
{ y } )
29 simpr 109 . . . . 5  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
y  C_  om )
30 simpl 108 . . . . . 6  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
y  e.  om )
3130snssd 3633 . . . . 5  |-  ( ( y  e.  om  /\  y  C_  om )  ->  { y }  C_  om )
3229, 31unssd 3220 . . . 4  |-  ( ( y  e.  om  /\  y  C_  om )  -> 
( y  u.  {
y } )  C_  om )
3328, 32eqsstrid 3111 . . 3  |-  ( ( y  e.  om  /\  y  C_  om )  ->  suc  y  C_  om )
3433ex 114 . 2  |-  ( y  e.  om  ->  (
y  C_  om  ->  suc  y  C_  om )
)
3527, 34mprg 2464 1  |-  ( A  e.  om  ->  A  C_ 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391    u. cun 3037    C_ wss 3039   (/)c0 3331   {csn 3495   suc csuc 4255   omcom 4472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12845  ax-bdor 12848  ax-bdal 12850  ax-bdex 12851  ax-bdeq 12852  ax-bdel 12853  ax-bdsb 12854  ax-bdsep 12916  ax-infvn 12973
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12873  df-bj-ind 12959
This theorem is referenced by:  bj-omtrans2  12989  bj-nnord  12990  bj-nn0suc  12996
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