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Theorem elomssom 4589
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 4590. (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 3170 . 2  |-  ( y  =  (/)  ->  ( y 
C_  om  <->  (/)  C_  om )
)
2 sseq1 3170 . 2  |-  ( y  =  x  ->  (
y  C_  om  <->  x  C_  om )
)
3 sseq1 3170 . 2  |-  ( y  =  suc  x  -> 
( y  C_  om  <->  suc  x  C_  om ) )
4 sseq1 3170 . 2  |-  ( y  =  A  ->  (
y  C_  om  <->  A  C_  om )
)
5 0ss 3453 . 2  |-  (/)  C_  om
6 unss 3301 . . . . 5  |-  ( ( x  C_  om  /\  {
x }  C_  om )  <->  ( x  u.  { x } )  C_  om )
7 vex 2733 . . . . . . 7  |-  x  e. 
_V
87snss 3709 . . . . . 6  |-  ( x  e.  om  <->  { x }  C_  om )
98anbi2i 454 . . . . 5  |-  ( ( x  C_  om  /\  x  e.  om )  <->  ( x  C_ 
om  /\  { x }  C_  om ) )
10 df-suc 4356 . . . . . 6  |-  suc  x  =  ( x  u. 
{ x } )
1110sseq1i 3173 . . . . 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 4584 1  |-  ( A  e.  om  ->  A  C_ 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2141    u. cun 3119    C_ wss 3121   (/)c0 3414   {csn 3583   suc csuc 4350   omcom 4574
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575
This theorem is referenced by:  elnn  4590  2ssom  6503  nninfwlpoimlemginf  7152
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