Theorem wetriext 4301

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

Step	Hyp	Ref	Expression
1		wetriext.b	. . . . 5 |- ( ph -> B e. A )
2		wetriext.ext	. . . . 5 |- ( ph -> A. z e. A ( z R B <-> z R C ) )
3		breq1 3767	. . . . . . 7 |- ( z = B -> ( z R B <-> B R B ) )
4		breq1 3767	. . . . . . 7 |- ( z = B -> ( z R C <-> B R C ) )
5	3, 4	bibi12d 224	. . . . . 6 |- ( z = B -> ( ( z R B <-> z R C ) <-> ( B R B <-> B R C ) ) )
6	5	rspcv 2652	. . . . 5 |- ( B e. A -> ( A. z e. A ( z R B <-> z R C ) -> ( B R B <-> B R C ) ) )
7	1, 2, 6	sylc 56	. . . 4 |- ( ph -> ( B R B <-> B R C ) )
8	7	biimpar 281	. . 3 |- ( ( ph /\ B R C ) -> B R B )
9		wetriext.we	. . . . . 6 |- ( ph -> R We A )
10		wefr 4095	. . . . . 6 |- ( R We A -> R Fr A )
11	9, 10	syl 14	. . . . 5 |- ( ph -> R Fr A )
12		wetriext.a	. . . . 5 |- ( ph -> A e. V )
13		frirrg 4087	. . . . 5 |- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B )
14	11, 12, 1, 13	syl3anc 1135	. . . 4 |- ( ph -> -. B R B )
15	14	adantr 261	. . 3 |- ( ( ph /\ B R C ) -> -. B R B )
16	8, 15	pm2.21dd 550	. 2 |- ( ( ph /\ B R C ) -> B = C )
17		simpr 103	. 2 |- ( ( ph /\ B = C ) -> B = C )
18		wetriext.c	. . . . 5 |- ( ph -> C e. A )
19		breq1 3767	. . . . . . 7 |- ( z = C -> ( z R B <-> C R B ) )
20		breq1 3767	. . . . . . 7 |- ( z = C -> ( z R C <-> C R C ) )
21	19, 20	bibi12d 224	. . . . . 6 |- ( z = C -> ( ( z R B <-> z R C ) <-> ( C R B <-> C R C ) ) )
22	21	rspcv 2652	. . . . 5 |- ( C e. A -> ( A. z e. A ( z R B <-> z R C ) -> ( C R B <-> C R C ) ) )
23	18, 2, 22	sylc 56	. . . 4 |- ( ph -> ( C R B <-> C R C ) )
24	23	biimpa 280	. . 3 |- ( ( ph /\ C R B ) -> C R C )
25		frirrg 4087	. . . . 5 |- ( ( R Fr A /\ A e. V /\ C e. A ) -> -. C R C )
26	11, 12, 18, 25	syl3anc 1135	. . . 4 |- ( ph -> -. C R C )
27	26	adantr 261	. . 3 |- ( ( ph /\ C R B ) -> -. C R C )
28	24, 27	pm2.21dd 550	. 2 |- ( ( ph /\ C R B ) -> B = C )
29		wetriext.tri	. . 3 |- ( ph -> A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) )
30		breq1 3767	. . . . . 6 |- ( a = B -> ( a R b <-> B R b ) )
31		eqeq1 2046	. . . . . 6 |- ( a = B -> ( a = b <-> B = b ) )
32		breq2 3768	. . . . . 6 |- ( a = B -> ( b R a <-> b R B ) )
33	30, 31, 32	3orbi123d 1206	. . . . 5 |- ( a = B -> ( ( a R b \/ a = b \/ b R a ) <-> ( B R b \/ B = b \/ b R B ) ) )
34		breq2 3768	. . . . . 6 |- ( b = C -> ( B R b <-> B R C ) )
35		eqeq2 2049	. . . . . 6 |- ( b = C -> ( B = b <-> B = C ) )
36		breq1 3767	. . . . . 6 |- ( b = C -> ( b R B <-> C R B ) )
37	34, 35, 36	3orbi123d 1206	. . . . 5 |- ( b = C -> ( ( B R b \/ B = b \/ b R B ) <-> ( B R C \/ B = C \/ C R B ) ) )
38	33, 37	rspc2v 2662	. . . 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 ) ) )
39	1, 18, 38	syl2anc 391	. . 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 ) ) )
40	29, 39	mpd 13	. 2 |- ( ph -> ( B R C \/ B = C \/ C R B ) )
41	16, 17, 28, 40	mpjao3dan 1202	1 |- ( ph -> B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ w3o 884   = wceq 1243   e. wcel 1393   A.wral 2306   class class class wbr 3764   Fr wfr 4065   We wwe 4067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-frfor 4068  df-frind 4069  df-wetr 4071

This theorem is referenced by: (None)