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Theorem nntri3or 6493
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 2241 . . . . 5  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
2 eqeq2 2187 . . . . 5  |-  ( x  =  B  ->  ( A  =  x  <->  A  =  B ) )
3 eleq1 2240 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
41, 2, 33orbi123d 1311 . . . 4  |-  ( x  =  B  ->  (
( A  e.  x  \/  A  =  x  \/  x  e.  A
)  <->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) ) )
54imbi2d 230 . . 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 2241 . . . . 5  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
7 eqeq2 2187 . . . . 5  |-  ( x  =  (/)  ->  ( A  =  x  <->  A  =  (/) ) )
8 eleq1 2240 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  A  <->  (/)  e.  A
) )
96, 7, 83orbi123d 1311 . . . 4  |-  ( x  =  (/)  ->  ( ( A  e.  x  \/  A  =  x  \/  x  e.  A )  <-> 
( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
) ) )
10 eleq2 2241 . . . . 5  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
11 eqeq2 2187 . . . . 5  |-  ( x  =  y  ->  ( A  =  x  <->  A  =  y ) )
12 eleq1 2240 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
1310, 11, 123orbi123d 1311 . . . 4  |-  ( x  =  y  ->  (
( A  e.  x  \/  A  =  x  \/  x  e.  A
)  <->  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) ) )
14 eleq2 2241 . . . . 5  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
15 eqeq2 2187 . . . . 5  |-  ( x  =  suc  y  -> 
( A  =  x  <-> 
A  =  suc  y
) )
16 eleq1 2240 . . . . 5  |-  ( x  =  suc  y  -> 
( x  e.  A  <->  suc  y  e.  A ) )
1714, 15, 163orbi123d 1311 . . . 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 4618 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
19 olc 711 . . . . . 6  |-  ( ( A  =  (/)  \/  (/)  e.  A
)  ->  ( A  e.  (/)  \/  ( A  =  (/)  \/  (/)  e.  A
) ) )
20 3orass 981 . . . . . 6  |-  ( ( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
)  <->  ( A  e.  (/)  \/  ( A  =  (/)  \/  (/)  e.  A ) ) )
2119, 20sylibr 134 . . . . 5  |-  ( ( A  =  (/)  \/  (/)  e.  A
)  ->  ( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A ) )
2218, 21syl 14 . . . 4  |-  ( A  e.  om  ->  ( A  e.  (/)  \/  A  =  (/)  \/  (/)  e.  A
) )
23 df-3or 979 . . . . . 6  |-  ( ( A  e.  y  \/  A  =  y  \/  y  e.  A )  <-> 
( ( A  e.  y  \/  A  =  y )  \/  y  e.  A ) )
24 elex 2748 . . . . . . . 8  |-  ( y  e.  om  ->  y  e.  _V )
25 elsuc2g 4405 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) ) )
26 3mix1 1166 . . . . . . . . 9  |-  ( A  e.  suc  y  -> 
( A  e.  suc  y  \/  A  =  suc  y  \/  suc  y  e.  A )
)
2725, 26syl6bir 164 . . . . . . . 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 6491 . . . . . . . . 9  |-  ( A  e.  om  ->  (
y  e.  A  <->  suc  y  e. 
suc  A ) )
30 elsuci 4403 . . . . . . . . 9  |-  ( suc  y  e.  suc  A  ->  ( suc  y  e.  A  \/  suc  y  =  A ) )
3129, 30syl6bi 163 . . . . . . . 8  |-  ( A  e.  om  ->  (
y  e.  A  -> 
( suc  y  e.  A  \/  suc  y  =  A ) ) )
32 eqcom 2179 . . . . . . . . . . . . 13  |-  ( suc  y  =  A  <->  A  =  suc  y )
3332orbi2i 762 . . . . . . . . . . . 12  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  <->  ( suc  y  e.  A  \/  A  =  suc  y ) )
3433biimpi 120 . . . . . . . . . . 11  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( suc  y  e.  A  \/  A  =  suc  y ) )
3534orcomd 729 . . . . . . . . . 10  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  =  suc  y  \/  suc  y  e.  A )
)
3635olcd 734 . . . . . . . . 9  |-  ( ( suc  y  e.  A  \/  suc  y  =  A )  ->  ( A  e.  suc  y  \/  ( A  =  suc  y  \/ 
suc  y  e.  A
) ) )
37 3orass 981 . . . . . . . . 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 134 . . . . . . . 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 719 . . . . . 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, 40biimtrid 152 . . . . 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 115 . . . 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 4600 . . 3  |-  ( x  e.  om  ->  ( A  e.  om  ->  ( A  e.  x  \/  A  =  x  \/  x  e.  A ) ) )
445, 43vtoclga 2803 . 2  |-  ( B  e.  om  ->  ( A  e.  om  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
) )
4544impcom 125 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 104    \/ wo 708    \/ w3o 977    = wceq 1353    e. wcel 2148   _Vcvv 2737   (/)c0 3422   suc csuc 4365   omcom 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590
This theorem is referenced by:  nntri2  6494  nntri1  6496  nntri3  6497  nntri2or2  6498  nndceq  6499  nndcel  6500  nnsseleq  6501  nntr2  6503  nnawordex  6529  nnwetri  6914  nnnninfeq  7125  ltsopi  7318  pitri3or  7320  frec2uzlt2d  10403  ennnfonelemk  12400  ennnfonelemex  12414
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