Theorem eqord1 8372

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 521	. . . . . 6 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> C e. S )
2		elisset 2735	. . . . . 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 274	. . . 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 275	. . . . 5 |- ( ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) /\ x = C ) -> A = M )
7		eqeq2 2174	. . . . . . . 8 |- ( C = D -> ( x = C <-> x = D ) )
8	7	adantl 275	. . . . . . 7 |- ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) -> ( x = C <-> x = D ) )
9	8	biimpa 294	. . . . . 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 2199	. . . 4 |- ( ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) /\ x = C ) -> M = N )
13	4, 12	exlimddv 1885	. . 3 |- ( ( ( ph /\ ( C e. S /\ D e. S ) ) /\ C = D ) -> M = N )
14	13	ex 114	. 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 8371	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )
20	19	con3d 621	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. M < N -> -. C < D ) )
21	15, 10, 5, 16, 17, 18	ltordlem 8371	. . . . . 6 |- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( D < C -> N < M ) )
22	21	con3d 621	. . . . 5 |- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( -. N < M -> -. D < C ) )
23	22	ancom2s 556	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. N < M -> -. D < C ) )
24	20, 23	anim12d 333	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( ( -. M < N /\ -. N < M ) -> ( -. C < D /\ -. D < C ) ) )
25	17	ralrimiva 2537	. . . . . 6 |- ( ph -> A. x e. S A e. RR )
26	5	eleq1d 2233	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
27	26	rspccva 2824	. . . . . 6 |- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR )
28	25, 27	sylan 281	. . . . 5 |- ( ( ph /\ C e. S ) -> M e. RR )
29	28	adantrr 471	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. RR )
30	10	eleq1d 2233	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
31	30	rspccva 2824	. . . . . 6 |- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR )
32	25, 31	sylan 281	. . . . 5 |- ( ( ph /\ D e. S ) -> N e. RR )
33	32	adantrl 470	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. RR )
34	29, 33	lttri3d 8004	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M = N <-> ( -. M < N /\ -. N < M ) ) )
35	16, 1	sseldi 3135	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> C e. RR )
36		simprr 522	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> D e. S )
37	16, 36	sseldi 3135	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> D e. RR )
38	35, 37	lttri3d 8004	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> ( -. C < D /\ -. D < C ) ) )
39	24, 34, 38	3imtr4d 202	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M = N -> C = D ) )
40	14, 39	impbid 128	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   E.wex 1479   e. wcel 2135   A.wral 2442   C_ wss 3111   class class class wbr 3976   RRcr 7743   < clt 7924

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-pre-ltirr 7856  ax-pre-apti 7859

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-xp 4604  df-pnf 7926  df-mnf 7927  df-ltxr 7929

This theorem is referenced by:  eqord2  8373  reef11  11626  nninfdclemf1  12324