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Theorem elomssom 4565
Description: A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4566. (Revised by BJ, 7-Aug-2024.)
Assertion
Ref Expression
elomssom  |-  ( A  e.  om  ->  A  C_ 
om )

Proof of Theorem elomssom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3151 . 2  |-  ( y  =  (/)  ->  ( y 
C_  om  <->  (/)  C_  om )
)
2 sseq1 3151 . 2  |-  ( y  =  x  ->  (
y  C_  om  <->  x  C_  om )
)
3 sseq1 3151 . 2  |-  ( y  =  suc  x  -> 
( y  C_  om  <->  suc  x  C_  om ) )
4 sseq1 3151 . 2  |-  ( y  =  A  ->  (
y  C_  om  <->  A  C_  om )
)
5 0ss 3432 . 2  |-  (/)  C_  om
6 unss 3281 . . . . 5  |-  ( ( x  C_  om  /\  {
x }  C_  om )  <->  ( x  u.  { x } )  C_  om )
7 vex 2715 . . . . . . 7  |-  x  e. 
_V
87snss 3686 . . . . . 6  |-  ( x  e.  om  <->  { x }  C_  om )
98anbi2i 453 . . . . 5  |-  ( ( x  C_  om  /\  x  e.  om )  <->  ( x  C_ 
om  /\  { x }  C_  om ) )
10 df-suc 4332 . . . . . 6  |-  suc  x  =  ( x  u. 
{ x } )
1110sseq1i 3154 . . . . 5  |-  ( suc  x  C_  om  <->  ( x  u.  { x } ) 
C_  om )
126, 9, 113bitr4i 211 . . . 4  |-  ( ( x  C_  om  /\  x  e.  om )  <->  suc  x  C_  om )
1312biimpi 119 . . 3  |-  ( ( x  C_  om  /\  x  e.  om )  ->  suc  x  C_  om )
1413expcom 115 . 2  |-  ( x  e.  om  ->  (
x  C_  om  ->  suc  x  C_  om )
)
151, 2, 3, 4, 5, 14finds 4560 1  |-  ( A  e.  om  ->  A  C_ 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2128    u. cun 3100    C_ wss 3102   (/)c0 3394   {csn 3560   suc csuc 4326   omcom 4550
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3774  df-int 3809  df-suc 4332  df-iom 4551
This theorem is referenced by:  elnn  4566  2ssom  13419
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