Theorem eqord1 8435

Description: A strictly increasing real function on a subset of RR is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)

Hypotheses

Ref	Expression
ltord.1	|- ( x = y -> A = B )
ltord.2	|- ( x = C -> A = M )
ltord.3	|- ( x = D -> A = N )
ltord.4	|- S C_ RR
ltord.5	|- ( ( ph /\ x e. S ) -> A e. RR )
ltord.6	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )

Assertion

Ref	Expression
eqord1	|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )

Distinct variable groups:   x, B   x, y, C   x, D, y   x, M, y   x, N, y   ph, x, y   x, S, y

Allowed substitution hints:   A( x, y)   B( y)

Proof of Theorem eqord1

Step	Hyp	Ref	Expression
1		simprl 529	. . . . . 6 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> C e. S )
2		elisset 2751	. . . . . 6 |- ( C e. S -> E. x x = C )
3	1, 2	syl 14	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> E. x x = C )
4	3	adantr 276	. . . 4 |- ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) -> E. x x = C )
5		ltord.2	. . . . . 6 |- ( x = C -> A = M )
6	5	adantl 277	. . . . 5 |- ( ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) /\ x = C ) -> A = M )
7		eqeq2 2187	. . . . . . . 8 |- ( C = D -> ( x = C <-> x = D ) )
8	7	adantl 277	. . . . . . 7 |- ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) -> ( x = C <-> x = D ) )
9	8	biimpa 296	. . . . . 6 |- ( ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) /\ x = C ) -> x = D )
10		ltord.3	. . . . . 6 |- ( x = D -> A = N )
11	9, 10	syl 14	. . . . 5 |- ( ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) /\ x = C ) -> A = N )
12	6, 11	eqtr3d 2212	. . . 4 |- ( ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) /\ x = C ) -> M = N )
13	4, 12	exlimddv 1898	. . 3 |- ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) -> M = N )
14	13	ex 115	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D -> M = N ) )
15		ltord.1	. . . . . 6 |- ( x = y -> A = B )
16		ltord.4	. . . . . 6 |- S C_ RR
17		ltord.5	. . . . . 6 |- ( ( ph /\ x e. S ) -> A e. RR )
18		ltord.6	. . . . . 6 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )
19	15, 5, 10, 16, 17, 18	ltordlem 8434	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )
20	19	con3d 631	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. M < N -> -. C < D ) )
21	15, 10, 5, 16, 17, 18	ltordlem 8434	. . . . . 6 |- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( D < C -> N < M ) )
22	21	con3d 631	. . . . 5 |- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( -. N < M -> -. D < C ) )
23	22	ancom2s 566	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. N < M -> -. D < C ) )
24	20, 23	anim12d 335	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( ( -. M < N /\ -. N < M ) -> ( -. C < D /\ -. D < C ) ) )
25	17	ralrimiva 2550	. . . . . 6 |- ( ph -> A. x e. S A e. RR )
26	5	eleq1d 2246	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
27	26	rspccva 2840	. . . . . 6 |- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR )
28	25, 27	sylan 283	. . . . 5 |- ( ( ph /\ C e. S ) -> M e. RR )
29	28	adantrr 479	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. RR )
30	10	eleq1d 2246	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
31	30	rspccva 2840	. . . . . 6 |- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR )
32	25, 31	sylan 283	. . . . 5 |- ( ( ph /\ D e. S ) -> N e. RR )
33	32	adantrl 478	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. RR )
34	29, 33	lttri3d 8067	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M = N <-> ( -. M < N /\ -. N < M ) ) )
35	16, 1	sselid 3153	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> C e. RR )
36		simprr 531	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> D e. S )
37	16, 36	sselid 3153	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> D e. RR )
38	35, 37	lttri3d 8067	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> ( -. C < D /\ -. D < C ) ) )
39	24, 34, 38	3imtr4d 203	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M = N -> C = D ) )
40	14, 39	impbid 129	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   C_ wss 3129   class class class wbr 4002   RRcr 7806   < clt 7987

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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-pre-ltirr 7919  ax-pre-apti 7922

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-pnf 7989  df-mnf 7990  df-ltxr 7992

This theorem is referenced by:  eqord2  8436  reef11  11699  nninfdclemf1  12444