Theorem wetriext 4570

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 4001	. . . . . 6 |- ( z = B -> ( z R B <-> B R B ) )
2		breq1 4001	. . . . . 6 |- ( z = B -> ( z R C <-> B R C ) )
3	1, 2	bibi12d 235	. . . . 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 2844	. . . 4 |- ( ph -> ( B R B <-> B R C ) )
7	6	biimpar 297	. . 3 |- ( ( ph /\ B R C ) -> B R B )
8		wetriext.we	. . . . . 6 |- ( ph -> R We A )
9		wefr 4352	. . . . . 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 4344	. . . . 5 |- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B )
13	10, 11, 5, 12	syl3anc 1238	. . . 4 |- ( ph -> -. B R B )
14	13	adantr 276	. . 3 |- ( ( ph /\ B R C ) -> -. B R B )
15	7, 14	pm2.21dd 620	. 2 |- ( ( ph /\ B R C ) -> B = C )
16		simpr 110	. 2 |- ( ( ph /\ B = C ) -> B = C )
17		breq1 4001	. . . . . 6 |- ( z = C -> ( z R B <-> C R B ) )
18		breq1 4001	. . . . . 6 |- ( z = C -> ( z R C <-> C R C ) )
19	17, 18	bibi12d 235	. . . . 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 2844	. . . 4 |- ( ph -> ( C R B <-> C R C ) )
22	21	biimpa 296	. . 3 |- ( ( ph /\ C R B ) -> C R C )
23		frirrg 4344	. . . . 5 |- ( ( R Fr A /\ A e. V /\ C e. A ) -> -. C R C )
24	10, 11, 20, 23	syl3anc 1238	. . . 4 |- ( ph -> -. C R C )
25	24	adantr 276	. . 3 |- ( ( ph /\ C R B ) -> -. C R C )
26	22, 25	pm2.21dd 620	. 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 4001	. . . . . 6 |- ( a = B -> ( a R b <-> B R b ) )
29		eqeq1 2182	. . . . . 6 |- ( a = B -> ( a = b <-> B = b ) )
30		breq2 4002	. . . . . 6 |- ( a = B -> ( b R a <-> b R B ) )
31	28, 29, 30	3orbi123d 1311	. . . . 5 |- ( a = B -> ( ( a R b \/ a = b \/ b R a ) <-> ( B R b \/ B = b \/ b R B ) ) )
32		breq2 4002	. . . . . 6 |- ( b = C -> ( B R b <-> B R C ) )
33		eqeq2 2185	. . . . . 6 |- ( b = C -> ( B = b <-> B = C ) )
34		breq1 4001	. . . . . 6 |- ( b = C -> ( b R B <-> C R B ) )
35	32, 33, 34	3orbi123d 1311	. . . . 5 |- ( b = C -> ( ( B R b \/ B = b \/ b R B ) <-> ( B R C \/ B = C \/ C R B ) ) )
36	31, 35	rspc2v 2852	. . . 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 411	. . 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 1307	1 |- ( ph -> B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 977   = wceq 1353   e. wcel 2146   A.wral 2453   class class class wbr 3998   Fr wfr 4322   We wwe 4324

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-frfor 4325  df-frind 4326  df-wetr 4328

This theorem is referenced by: (None)