Theorem wetriext 4554

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		breq1 3985	. . . . . 6 |- ( z = B -> ( z R B <-> B R B ) )
2		breq1 3985	. . . . . 6 |- ( z = B -> ( z R C <-> B R C ) )
3	1, 2	bibi12d 234	. . . . 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 )
6	3, 4, 5	rspcdva 2835	. . . 4 |- ( ph -> ( B R B <-> B R C ) )
7	6	biimpar 295	. . 3 |- ( ( ph /\ B R C ) -> B R B )
8		wetriext.we	. . . . . 6 |- ( ph -> R We A )
9		wefr 4336	. . . . . 6 |- ( R We A -> R Fr A )
10	8, 9	syl 14	. . . . 5 |- ( ph -> R Fr A )
11		wetriext.a	. . . . 5 |- ( ph -> A e. V )
12		frirrg 4328	. . . . 5 |- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B )
13	10, 11, 5, 12	syl3anc 1228	. . . 4 |- ( ph -> -. B R B )
14	13	adantr 274	. . 3 |- ( ( ph /\ B R C ) -> -. B R B )
15	7, 14	pm2.21dd 610	. 2 |- ( ( ph /\ B R C ) -> B = C )
16		simpr 109	. 2 |- ( ( ph /\ B = C ) -> B = C )
17		breq1 3985	. . . . . 6 |- ( z = C -> ( z R B <-> C R B ) )
18		breq1 3985	. . . . . 6 |- ( z = C -> ( z R C <-> C R C ) )
19	17, 18	bibi12d 234	. . . . 5 |- ( z = C -> ( ( z R B <-> z R C ) <-> ( C R B <-> C R C ) ) )
20		wetriext.c	. . . . 5 |- ( ph -> C e. A )
21	19, 4, 20	rspcdva 2835	. . . 4 |- ( ph -> ( C R B <-> C R C ) )
22	21	biimpa 294	. . 3 |- ( ( ph /\ C R B ) -> C R C )
23		frirrg 4328	. . . . 5 |- ( ( R Fr A /\ A e. V /\ C e. A ) -> -. C R C )
24	10, 11, 20, 23	syl3anc 1228	. . . 4 |- ( ph -> -. C R C )
25	24	adantr 274	. . 3 |- ( ( ph /\ C R B ) -> -. C R C )
26	22, 25	pm2.21dd 610	. 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 3985	. . . . . 6 |- ( a = B -> ( a R b <-> B R b ) )
29		eqeq1 2172	. . . . . 6 |- ( a = B -> ( a = b <-> B = b ) )
30		breq2 3986	. . . . . 6 |- ( a = B -> ( b R a <-> b R B ) )
31	28, 29, 30	3orbi123d 1301	. . . . 5 |- ( a = B -> ( ( a R b \/ a = b \/ b R a ) <-> ( B R b \/ B = b \/ b R B ) ) )
32		breq2 3986	. . . . . 6 |- ( b = C -> ( B R b <-> B R C ) )
33		eqeq2 2175	. . . . . 6 |- ( b = C -> ( B = b <-> B = C ) )
34		breq1 3985	. . . . . 6 |- ( b = C -> ( b R B <-> C R B ) )
35	32, 33, 34	3orbi123d 1301	. . . . 5 |- ( b = C -> ( ( B R b \/ B = b \/ b R B ) <-> ( B R C \/ B = C \/ C R B ) ) )
36	31, 35	rspc2v 2843	. . . 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 ) ) )
37	5, 20, 36	syl2anc 409	. . 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 ) ) )
38	27, 37	mpd 13	. 2 |- ( ph -> ( B R C \/ B = C \/ C R B ) )
39	15, 16, 26, 38	mpjao3dan 1297	1 |- ( ph -> B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 967   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982   Fr wfr 4306   We wwe 4308

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-frfor 4309  df-frind 4310  df-wetr 4312

This theorem is referenced by: (None)