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Theorem wetriext 4382
Description: A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
Hypotheses
Ref Expression
wetriext.we  |-  ( ph  ->  R  We  A )
wetriext.a  |-  ( ph  ->  A  e.  V )
wetriext.tri  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
wetriext.b  |-  ( ph  ->  B  e.  A )
wetriext.c  |-  ( ph  ->  C  e.  A )
wetriext.ext  |-  ( ph  ->  A. z  e.  A  ( z R B  <-> 
z R C ) )
Assertion
Ref Expression
wetriext  |-  ( ph  ->  B  =  C )
Distinct variable groups:    A, a, b   
z, A    B, a,
b    z, B    C, b    z, C    R, a, b    z, R
Allowed substitution hints:    ph( z, a, b)    C( a)    V( z, a, b)

Proof of Theorem wetriext
StepHypRef Expression
1 breq1 3840 . . . . . 6  |-  ( z  =  B  ->  (
z R B  <->  B R B ) )
2 breq1 3840 . . . . . 6  |-  ( z  =  B  ->  (
z R C  <->  B R C ) )
31, 2bibi12d 233 . . . . 5  |-  ( z  =  B  ->  (
( z R B  <-> 
z R C )  <-> 
( B R B  <-> 
B R C ) ) )
4 wetriext.ext . . . . 5  |-  ( ph  ->  A. z  e.  A  ( z R B  <-> 
z R C ) )
5 wetriext.b . . . . 5  |-  ( ph  ->  B  e.  A )
63, 4, 5rspcdva 2727 . . . 4  |-  ( ph  ->  ( B R B  <-> 
B R C ) )
76biimpar 291 . . 3  |-  ( (
ph  /\  B R C )  ->  B R B )
8 wetriext.we . . . . . 6  |-  ( ph  ->  R  We  A )
9 wefr 4176 . . . . . 6  |-  ( R  We  A  ->  R  Fr  A )
108, 9syl 14 . . . . 5  |-  ( ph  ->  R  Fr  A )
11 wetriext.a . . . . 5  |-  ( ph  ->  A  e.  V )
12 frirrg 4168 . . . . 5  |-  ( ( R  Fr  A  /\  A  e.  V  /\  B  e.  A )  ->  -.  B R B )
1310, 11, 5, 12syl3anc 1174 . . . 4  |-  ( ph  ->  -.  B R B )
1413adantr 270 . . 3  |-  ( (
ph  /\  B R C )  ->  -.  B R B )
157, 14pm2.21dd 585 . 2  |-  ( (
ph  /\  B R C )  ->  B  =  C )
16 simpr 108 . 2  |-  ( (
ph  /\  B  =  C )  ->  B  =  C )
17 breq1 3840 . . . . . 6  |-  ( z  =  C  ->  (
z R B  <->  C R B ) )
18 breq1 3840 . . . . . 6  |-  ( z  =  C  ->  (
z R C  <->  C R C ) )
1917, 18bibi12d 233 . . . . 5  |-  ( z  =  C  ->  (
( z R B  <-> 
z R C )  <-> 
( C R B  <-> 
C R C ) ) )
20 wetriext.c . . . . 5  |-  ( ph  ->  C  e.  A )
2119, 4, 20rspcdva 2727 . . . 4  |-  ( ph  ->  ( C R B  <-> 
C R C ) )
2221biimpa 290 . . 3  |-  ( (
ph  /\  C R B )  ->  C R C )
23 frirrg 4168 . . . . 5  |-  ( ( R  Fr  A  /\  A  e.  V  /\  C  e.  A )  ->  -.  C R C )
2410, 11, 20, 23syl3anc 1174 . . . 4  |-  ( ph  ->  -.  C R C )
2524adantr 270 . . 3  |-  ( (
ph  /\  C R B )  ->  -.  C R C )
2622, 25pm2.21dd 585 . 2  |-  ( (
ph  /\  C R B )  ->  B  =  C )
27 wetriext.tri . . 3  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
28 breq1 3840 . . . . . 6  |-  ( a  =  B  ->  (
a R b  <->  B R
b ) )
29 eqeq1 2094 . . . . . 6  |-  ( a  =  B  ->  (
a  =  b  <->  B  =  b ) )
30 breq2 3841 . . . . . 6  |-  ( a  =  B  ->  (
b R a  <->  b R B ) )
3128, 29, 303orbi123d 1247 . . . . 5  |-  ( a  =  B  ->  (
( a R b  \/  a  =  b  \/  b R a )  <->  ( B R b  \/  B  =  b  \/  b R B ) ) )
32 breq2 3841 . . . . . 6  |-  ( b  =  C  ->  ( B R b  <->  B R C ) )
33 eqeq2 2097 . . . . . 6  |-  ( b  =  C  ->  ( B  =  b  <->  B  =  C ) )
34 breq1 3840 . . . . . 6  |-  ( b  =  C  ->  (
b R B  <->  C R B ) )
3532, 33, 343orbi123d 1247 . . . . 5  |-  ( b  =  C  ->  (
( B R b  \/  B  =  b  \/  b R B )  <->  ( B R C  \/  B  =  C  \/  C R B ) ) )
3631, 35rspc2v 2733 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )  ->  ( B R C  \/  B  =  C  \/  C R B ) ) )
375, 20, 36syl2anc 403 . . 3  |-  ( ph  ->  ( A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )  ->  ( B R C  \/  B  =  C  \/  C R B ) ) )
3827, 37mpd 13 . 2  |-  ( ph  ->  ( B R C  \/  B  =  C  \/  C R B ) )
3915, 16, 26, 38mpjao3dan 1243 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ w3o 923    = wceq 1289    e. wcel 1438   A.wral 2359   class class class wbr 3837    Fr wfr 4146    We wwe 4148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-frfor 4149  df-frind 4150  df-wetr 4152
This theorem is referenced by: (None)
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