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Theorem 0elnn 4618
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 2184 . . 3  |-  ( x  =  (/)  ->  ( x  =  (/)  <->  (/)  =  (/) ) )
2 eleq2 2241 . . 3  |-  ( x  =  (/)  ->  ( (/)  e.  x  <->  (/)  e.  (/) ) )
31, 2orbi12d 793 . 2  |-  ( x  =  (/)  ->  ( ( x  =  (/)  \/  (/)  e.  x
)  <->  ( (/)  =  (/)  \/  (/)  e.  (/) ) ) )
4 eqeq1 2184 . . 3  |-  ( x  =  y  ->  (
x  =  (/)  <->  y  =  (/) ) )
5 eleq2 2241 . . 3  |-  ( x  =  y  ->  ( (/) 
e.  x  <->  (/)  e.  y ) )
64, 5orbi12d 793 . 2  |-  ( x  =  y  ->  (
( x  =  (/)  \/  (/)  e.  x )  <->  ( y  =  (/)  \/  (/)  e.  y ) ) )
7 eqeq1 2184 . . 3  |-  ( x  =  suc  y  -> 
( x  =  (/)  <->  suc  y  =  (/) ) )
8 eleq2 2241 . . 3  |-  ( x  =  suc  y  -> 
( (/)  e.  x  <->  (/)  e.  suc  y ) )
97, 8orbi12d 793 . 2  |-  ( x  =  suc  y  -> 
( ( x  =  (/)  \/  (/)  e.  x )  <-> 
( suc  y  =  (/) 
\/  (/)  e.  suc  y
) ) )
10 eqeq1 2184 . . 3  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
11 eleq2 2241 . . 3  |-  ( x  =  A  ->  ( (/) 
e.  x  <->  (/)  e.  A
) )
1210, 11orbi12d 793 . 2  |-  ( x  =  A  ->  (
( x  =  (/)  \/  (/)  e.  x )  <->  ( A  =  (/)  \/  (/)  e.  A
) ) )
13 eqid 2177 . . 3  |-  (/)  =  (/)
1413orci 731 . 2  |-  ( (/)  =  (/)  \/  (/)  e.  (/) )
15 0ex 4130 . . . . . . 7  |-  (/)  e.  _V
1615sucid 4417 . . . . . 6  |-  (/)  e.  suc  (/)
17 suceq 4402 . . . . . 6  |-  ( y  =  (/)  ->  suc  y  =  suc  (/) )
1816, 17eleqtrrid 2267 . . . . 5  |-  ( y  =  (/)  ->  (/)  e.  suc  y )
1918a1i 9 . . . 4  |-  ( y  e.  om  ->  (
y  =  (/)  ->  (/)  e.  suc  y ) )
20 sssucid 4415 . . . . . 6  |-  y  C_  suc  y
2120a1i 9 . . . . 5  |-  ( y  e.  om  ->  y  C_ 
suc  y )
2221sseld 3154 . . . 4  |-  ( y  e.  om  ->  ( (/) 
e.  y  ->  (/)  e.  suc  y ) )
2319, 22jaod 717 . . 3  |-  ( y  e.  om  ->  (
( y  =  (/)  \/  (/)  e.  y )  ->  (/) 
e.  suc  y )
)
24 olc 711 . . 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 4599 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 708    = wceq 1353    e. wcel 2148    C_ wss 3129   (/)c0 3422   suc csuc 4365   omcom 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590
This theorem is referenced by:  nn0eln0  4619  nnsucsssuc  6492  nntri3or  6493  nnm00  6530  ssfilem  6874  diffitest  6886  fiintim  6927  enumct  7113  nnnninfeq  7125  elni2  7312  enq0tr  7432  bj-charfunr  14482
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