ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nntri3or Unicode version

Theorem nntri3or 6355
Description: Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
Assertion
Ref Expression
nntri3or  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )

Proof of Theorem nntri3or
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2179 . . . . 5  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
2 eqeq2 2125 . . . . 5  |-  ( x  =  B  ->  ( A  =  x  <->  A  =  B ) )
3 eleq1 2178 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
41, 2, 33orbi123d 1272 . . . 4  |-  ( x  =  B  ->  (
( A  e.  x  \/  A  =  x  \/  x  e.  A
)  <->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) ) )
54imbi2d 229 . . 3  |-  ( x  =  B  ->  (
( A  e.  om  ->  ( A  e.  x  \/  A  =  x  \/  x  e.  A
) )  <->  ( A  e.  om  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) ) ) )
6 eleq2 2179 . . . . 5  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
7 eqeq2 2125 . . . . 5  |-  ( x  =  (/)  ->  ( A  =  x  <->  A  =  (/) ) )
8 eleq1 2178 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  A  <->  (/)  e.  A
) )
96, 7, 83orbi123d 1272 . . . 4  |-  ( x  =  (/)  ->  ( ( A  e.  x  \/  A  =  x  \/  x  e.  A )  <-> 
( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
) ) )
10 eleq2 2179 . . . . 5  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
11 eqeq2 2125 . . . . 5  |-  ( x  =  y  ->  ( A  =  x  <->  A  =  y ) )
12 eleq1 2178 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
1310, 11, 123orbi123d 1272 . . . 4  |-  ( x  =  y  ->  (
( A  e.  x  \/  A  =  x  \/  x  e.  A
)  <->  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) ) )
14 eleq2 2179 . . . . 5  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
15 eqeq2 2125 . . . . 5  |-  ( x  =  suc  y  -> 
( A  =  x  <-> 
A  =  suc  y
) )
16 eleq1 2178 . . . . 5  |-  ( x  =  suc  y  -> 
( x  e.  A  <->  suc  y  e.  A ) )
1714, 15, 163orbi123d 1272 . . . 4  |-  ( x  =  suc  y  -> 
( ( A  e.  x  \/  A  =  x  \/  x  e.  A )  <->  ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
) )
18 0elnn 4500 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
19 olc 683 . . . . . 6  |-  ( ( A  =  (/)  \/  (/)  e.  A
)  ->  ( A  e.  (/)  \/  ( A  =  (/)  \/  (/)  e.  A
) ) )
20 3orass 948 . . . . . 6  |-  ( ( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
)  <->  ( A  e.  (/)  \/  ( A  =  (/)  \/  (/)  e.  A ) ) )
2119, 20sylibr 133 . . . . 5  |-  ( ( A  =  (/)  \/  (/)  e.  A
)  ->  ( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A ) )
2218, 21syl 14 . . . 4  |-  ( A  e.  om  ->  ( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
) )
23 df-3or 946 . . . . . 6  |-  ( ( A  e.  y  \/  A  =  y  \/  y  e.  A )  <-> 
( ( A  e.  y  \/  A  =  y )  \/  y  e.  A ) )
24 elex 2669 . . . . . . . 8  |-  ( y  e.  om  ->  y  e.  _V )
25 elsuc2g 4295 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) ) )
26 3mix1 1133 . . . . . . . . 9  |-  ( A  e.  suc  y  -> 
( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
)
2725, 26syl6bir 163 . . . . . . . 8  |-  ( y  e.  _V  ->  (
( A  e.  y  \/  A  =  y )  ->  ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
) )
2824, 27syl 14 . . . . . . 7  |-  ( y  e.  om  ->  (
( A  e.  y  \/  A  =  y )  ->  ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
) )
29 nnsucelsuc 6353 . . . . . . . . 9  |-  ( A  e.  om  ->  (
y  e.  A  <->  suc  y  e. 
suc  A ) )
30 elsuci 4293 . . . . . . . . 9  |-  ( suc  y  e.  suc  A  ->  ( suc  y  e.  A  \/  suc  y  =  A ) )
3129, 30syl6bi 162 . . . . . . . 8  |-  ( A  e.  om  ->  (
y  e.  A  -> 
( suc  y  e.  A  \/  suc  y  =  A ) ) )
32 eqcom 2117 . . . . . . . . . . . . 13  |-  ( suc  y  =  A  <->  A  =  suc  y )
3332orbi2i 734 . . . . . . . . . . . 12  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  <->  ( suc  y  e.  A  \/  A  =  suc  y ) )
3433biimpi 119 . . . . . . . . . . 11  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( suc  y  e.  A  \/  A  =  suc  y ) )
3534orcomd 701 . . . . . . . . . 10  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  =  suc  y  \/  suc  y  e.  A )
)
3635olcd 706 . . . . . . . . 9  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  e.  suc  y  \/  ( A  =  suc  y  \/ 
suc  y  e.  A
) ) )
37 3orass 948 . . . . . . . . 9  |-  ( ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )  <->  ( A  e.  suc  y  \/  ( A  =  suc  y  \/ 
suc  y  e.  A
) ) )
3836, 37sylibr 133 . . . . . . . 8  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
)
3931, 38syl6 33 . . . . . . 7  |-  ( A  e.  om  ->  (
y  e.  A  -> 
( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
) )
4028, 39jaao 691 . . . . . 6  |-  ( ( y  e.  om  /\  A  e.  om )  ->  ( ( ( A  e.  y  \/  A  =  y )  \/  y  e.  A )  ->  ( A  e. 
suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
) )
4123, 40syl5bi 151 . . . . 5  |-  ( ( y  e.  om  /\  A  e.  om )  ->  ( ( A  e.  y  \/  A  =  y  \/  y  e.  A )  ->  ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A
) ) )
4241ex 114 . . . 4  |-  ( y  e.  om  ->  ( A  e.  om  ->  ( ( A  e.  y  \/  A  =  y  \/  y  e.  A
)  ->  ( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
) ) )
439, 13, 17, 22, 42finds2 4483 . . 3  |-  ( x  e.  om  ->  ( A  e.  om  ->  ( A  e.  x  \/  A  =  x  \/  x  e.  A ) ) )
445, 43vtoclga 2724 . 2  |-  ( B  e.  om  ->  ( A  e.  om  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
) )
4544impcom 124 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    \/ w3o 944    = wceq 1314    e. wcel 1463   _Vcvv 2658   (/)c0 3331   suc csuc 4255   omcom 4472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473
This theorem is referenced by:  nntri2  6356  nntri1  6358  nntri3  6359  nntri2or2  6360  nndceq  6361  nndcel  6362  nnsseleq  6363  nntr2  6365  nnawordex  6390  nnwetri  6770  ltsopi  7092  pitri3or  7094  frec2uzlt2d  10128  ennnfonelemk  11819  ennnfonelemex  11833  nninfalllemn  13036
  Copyright terms: Public domain W3C validator