ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nndceq Unicode version

Theorem nndceq 6446
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 4577. (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 6440 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
2 elirr 4500 . . . . . . 7  |-  -.  A  e.  A
3 eleq2 2221 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
42, 3mtbii 664 . . . . . 6  |-  ( A  =  B  ->  -.  A  e.  B )
54con2i 617 . . . . 5  |-  ( A  e.  B  ->  -.  A  =  B )
65olcd 724 . . . 4  |-  ( A  e.  B  ->  ( A  =  B  \/  -.  A  =  B
) )
7 orc 702 . . . 4  |-  ( A  =  B  ->  ( A  =  B  \/  -.  A  =  B
) )
8 elirr 4500 . . . . . . 7  |-  -.  B  e.  B
9 eleq2 2221 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
108, 9mtbiri 665 . . . . . 6  |-  ( A  =  B  ->  -.  B  e.  A )
1110con2i 617 . . . . 5  |-  ( B  e.  A  ->  -.  A  =  B )
1211olcd 724 . . . 4  |-  ( B  e.  A  ->  ( A  =  B  \/  -.  A  =  B
) )
136, 7, 123jaoi 1285 . . 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 821 . 2  |-  (DECID  A  =  B  <->  ( A  =  B  \/  -.  A  =  B ) )
1614, 15sylibr 133 1  |-  ( ( A  e.  om  /\  B  e.  om )  -> DECID  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698  DECID wdc 820    \/ w3o 962    = wceq 1335    e. wcel 2128   omcom 4549
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 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  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-ne 2328  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 3773  df-int 3808  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550
This theorem is referenced by:  nndifsnid  6454  fidceq  6814  unsnfidcex  6864  unsnfidcel  6865  enqdc  7281  nninfsellemdc  13582
  Copyright terms: Public domain W3C validator