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Theorem nndceq 6478
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 4602. (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 6472 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
2 elirr 4525 . . . . . . 7  |-  -.  A  e.  A
3 eleq2 2234 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
42, 3mtbii 669 . . . . . 6  |-  ( A  =  B  ->  -.  A  e.  B )
54con2i 622 . . . . 5  |-  ( A  e.  B  ->  -.  A  =  B )
65olcd 729 . . . 4  |-  ( A  e.  B  ->  ( A  =  B  \/  -.  A  =  B
) )
7 orc 707 . . . 4  |-  ( A  =  B  ->  ( A  =  B  \/  -.  A  =  B
) )
8 elirr 4525 . . . . . . 7  |-  -.  B  e.  B
9 eleq2 2234 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
108, 9mtbiri 670 . . . . . 6  |-  ( A  =  B  ->  -.  B  e.  A )
1110con2i 622 . . . . 5  |-  ( B  e.  A  ->  -.  A  =  B )
1211olcd 729 . . . 4  |-  ( B  e.  A  ->  ( A  =  B  \/  -.  A  =  B
) )
136, 7, 123jaoi 1298 . . 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 830 . 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 703  DECID wdc 829    \/ w3o 972    = wceq 1348    e. wcel 2141   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-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  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-ne 2341  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-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575
This theorem is referenced by:  nndifsnid  6486  fidceq  6847  unsnfidcex  6897  unsnfidcel  6898  nninfwlporlemd  7148  nninfwlporlem  7149  nninfwlpoimlemg  7151  nninfwlpoimlemginf  7152  enqdc  7323  nninfsellemdc  14043
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