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Theorem nndceq 6143
Description: Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where  B is zero, see nndceq0 4365. (Contributed by Jim Kingdon, 31-Aug-2019.)
Assertion
Ref Expression
nndceq  |-  ( ( A  e.  om  /\  B  e.  om )  -> DECID  A  =  B )

Proof of Theorem nndceq
StepHypRef Expression
1 nntri3or 6137 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
2 elirr 4292 . . . . . . 7  |-  -.  A  e.  A
3 eleq2 2143 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
42, 3mtbii 632 . . . . . 6  |-  ( A  =  B  ->  -.  A  e.  B )
54con2i 590 . . . . 5  |-  ( A  e.  B  ->  -.  A  =  B )
65olcd 686 . . . 4  |-  ( A  e.  B  ->  ( A  =  B  \/  -.  A  =  B
) )
7 orc 666 . . . 4  |-  ( A  =  B  ->  ( A  =  B  \/  -.  A  =  B
) )
8 elirr 4292 . . . . . . 7  |-  -.  B  e.  B
9 eleq2 2143 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
108, 9mtbiri 633 . . . . . 6  |-  ( A  =  B  ->  -.  B  e.  A )
1110con2i 590 . . . . 5  |-  ( B  e.  A  ->  -.  A  =  B )
1211olcd 686 . . . 4  |-  ( B  e.  A  ->  ( A  =  B  \/  -.  A  =  B
) )
136, 7, 123jaoi 1235 . . 3  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  ->  ( A  =  B  \/  -.  A  =  B ) )
141, 13syl 14 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  \/  -.  A  =  B ) )
15 df-dc 777 . 2  |-  (DECID  A  =  B  <->  ( A  =  B  \/  -.  A  =  B ) )
1614, 15sylibr 132 1  |-  ( ( A  e.  om  /\  B  e.  om )  -> DECID  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662  DECID wdc 776    \/ w3o 919    = wceq 1285    e. wcel 1434   omcom 4339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340
This theorem is referenced by:  nndifsnid  6146  fidceq  6404  unsnfidcex  6440  unsnfidcel  6441  enqdc  6613
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