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Theorem 0elnn 4368
Description: A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
Assertion
Ref Expression
0elnn  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )

Proof of Theorem 0elnn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2062 . . 3  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
2 eleq2 2117 . . 3  |-  ( x  =  (/)  ->  ( (/)  e.  x  <->  (/)  e.  (/) ) )
31, 2orbi12d 717 . 2  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  (/)  e.  x
)  <->  ( (/)  =  (/)  \/  (/)  e.  (/) ) ) )
4 eqeq1 2062 . . 3  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
5 eleq2 2117 . . 3  |-  ( x  =  y  ->  ( (/) 
e.  x  <->  (/)  e.  y ) )
64, 5orbi12d 717 . 2  |-  ( x  =  y  ->  (
( x  =  (/)  \/  (/)  e.  x )  <->  ( y  =  (/)  \/  (/)  e.  y ) ) )
7 eqeq1 2062 . . 3  |-  ( x  =  suc  y  -> 
( x  =  (/)  <->  suc  y  =  (/) ) )
8 eleq2 2117 . . 3  |-  ( x  =  suc  y  -> 
( (/)  e.  x  <->  (/)  e.  suc  y ) )
97, 8orbi12d 717 . 2  |-  ( x  =  suc  y  -> 
( ( x  =  (/)  \/  (/)  e.  x )  <-> 
( suc  y  =  (/) 
\/  (/)  e.  suc  y
) ) )
10 eqeq1 2062 . . 3  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
11 eleq2 2117 . . 3  |-  ( x  =  A  ->  ( (/) 
e.  x  <->  (/)  e.  A
) )
1210, 11orbi12d 717 . 2  |-  ( x  =  A  ->  (
( x  =  (/)  \/  (/)  e.  x )  <->  ( A  =  (/)  \/  (/)  e.  A
) ) )
13 eqid 2056 . . 3  |-  (/)  =  (/)
1413orci 660 . 2  |-  ( (/)  =  (/)  \/  (/)  e.  (/) )
15 0ex 3912 . . . . . . 7  |-  (/)  e.  _V
1615sucid 4182 . . . . . 6  |-  (/)  e.  suc  (/)
17 suceq 4167 . . . . . 6  |-  ( y  =  (/)  ->  suc  y  =  suc  (/) )
1816, 17syl5eleqr 2143 . . . . 5  |-  ( y  =  (/)  ->  (/)  e.  suc  y )
1918a1i 9 . . . 4  |-  ( y  e.  om  ->  (
y  =  (/)  ->  (/)  e.  suc  y ) )
20 sssucid 4180 . . . . . 6  |-  y  C_  suc  y
2120a1i 9 . . . . 5  |-  ( y  e.  om  ->  y  C_ 
suc  y )
2221sseld 2972 . . . 4  |-  ( y  e.  om  ->  ( (/) 
e.  y  ->  (/)  e.  suc  y ) )
2319, 22jaod 647 . . 3  |-  ( y  e.  om  ->  (
( y  =  (/)  \/  (/)  e.  y )  ->  (/) 
e.  suc  y )
)
24 olc 642 . . 3  |-  ( (/)  e.  suc  y  ->  ( suc  y  =  (/)  \/  (/)  e.  suc  y ) )
2523, 24syl6 33 . 2  |-  ( y  e.  om  ->  (
( y  =  (/)  \/  (/)  e.  y )  -> 
( suc  y  =  (/) 
\/  (/)  e.  suc  y
) ) )
263, 6, 9, 12, 14, 25finds 4351 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639    = wceq 1259    e. wcel 1409    C_ wss 2945   (/)c0 3252   suc csuc 4130   omcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342
This theorem is referenced by:  nn0eln0  4369  nnsucsssuc  6102  nntri3or  6103  nnm00  6133  ssfiexmid  6367  diffitest  6375  elni2  6470  enq0tr  6590
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