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Theorem nntri3or 6357
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 2181 . . . . 5  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
2 eqeq2 2127 . . . . 5  |-  ( x  =  B  ->  ( A  =  x  <->  A  =  B ) )
3 eleq1 2180 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
41, 2, 33orbi123d 1274 . . . 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 2181 . . . . 5  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
7 eqeq2 2127 . . . . 5  |-  ( x  =  (/)  ->  ( A  =  x  <->  A  =  (/) ) )
8 eleq1 2180 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  A  <->  (/)  e.  A
) )
96, 7, 83orbi123d 1274 . . . 4  |-  ( x  =  (/)  ->  ( ( A  e.  x  \/  A  =  x  \/  x  e.  A )  <-> 
( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
) ) )
10 eleq2 2181 . . . . 5  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
11 eqeq2 2127 . . . . 5  |-  ( x  =  y  ->  ( A  =  x  <->  A  =  y ) )
12 eleq1 2180 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
1310, 11, 123orbi123d 1274 . . . 4  |-  ( x  =  y  ->  (
( A  e.  x  \/  A  =  x  \/  x  e.  A
)  <->  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) ) )
14 eleq2 2181 . . . . 5  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
15 eqeq2 2127 . . . . 5  |-  ( x  =  suc  y  -> 
( A  =  x  <-> 
A  =  suc  y
) )
16 eleq1 2180 . . . . 5  |-  ( x  =  suc  y  -> 
( x  e.  A  <->  suc  y  e.  A ) )
1714, 15, 163orbi123d 1274 . . . 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 4502 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
19 olc 685 . . . . . 6  |-  ( ( A  =  (/)  \/  (/)  e.  A
)  ->  ( A  e.  (/)  \/  ( A  =  (/)  \/  (/)  e.  A
) ) )
20 3orass 950 . . . . . 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 948 . . . . . 6  |-  ( ( A  e.  y  \/  A  =  y  \/  y  e.  A )  <-> 
( ( A  e.  y  \/  A  =  y )  \/  y  e.  A ) )
24 elex 2671 . . . . . . . 8  |-  ( y  e.  om  ->  y  e.  _V )
25 elsuc2g 4297 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) ) )
26 3mix1 1135 . . . . . . . . 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 6355 . . . . . . . . 9  |-  ( A  e.  om  ->  (
y  e.  A  <->  suc  y  e. 
suc  A ) )
30 elsuci 4295 . . . . . . . . 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 2119 . . . . . . . . . . . . 13  |-  ( suc  y  =  A  <->  A  =  suc  y )
3332orbi2i 736 . . . . . . . . . . . 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 703 . . . . . . . . . 10  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  =  suc  y  \/  suc  y  e.  A )
)
3635olcd 708 . . . . . . . . 9  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  e.  suc  y  \/  ( A  =  suc  y  \/ 
suc  y  e.  A
) ) )
37 3orass 950 . . . . . . . . 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 693 . . . . . 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 4485 . . 3  |-  ( x  e.  om  ->  ( A  e.  om  ->  ( A  e.  x  \/  A  =  x  \/  x  e.  A ) ) )
445, 43vtoclga 2726 . 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 682    \/ w3o 946    = wceq 1316    e. wcel 1465   _Vcvv 2660   (/)c0 3333   suc csuc 4257   omcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475
This theorem is referenced by:  nntri2  6358  nntri1  6360  nntri3  6361  nntri2or2  6362  nndceq  6363  nndcel  6364  nnsseleq  6365  nntr2  6367  nnawordex  6392  nnwetri  6772  ltsopi  7096  pitri3or  7098  frec2uzlt2d  10132  ennnfonelemk  11824  ennnfonelemex  11838  nninfalllemn  13098
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