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Mirrors > Home > ILE Home > Th. List > eqord1 | Unicode version |
Description: A strictly increasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.) |
Ref | Expression |
---|---|
ltord.1 | |
ltord.2 | |
ltord.3 | |
ltord.4 | |
ltord.5 | |
ltord.6 |
Ref | Expression |
---|---|
eqord1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 520 | . . . . . 6 | |
2 | elisset 2695 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | ltord.2 | . . . . . 6 | |
6 | 5 | adantl 275 | . . . . 5 |
7 | eqeq2 2147 | . . . . . . . 8 | |
8 | 7 | adantl 275 | . . . . . . 7 |
9 | 8 | biimpa 294 | . . . . . 6 |
10 | ltord.3 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 6, 11 | eqtr3d 2172 | . . . 4 |
13 | 4, 12 | exlimddv 1870 | . . 3 |
14 | 13 | ex 114 | . 2 |
15 | ltord.1 | . . . . . 6 | |
16 | ltord.4 | . . . . . 6 | |
17 | ltord.5 | . . . . . 6 | |
18 | ltord.6 | . . . . . 6 | |
19 | 15, 5, 10, 16, 17, 18 | ltordlem 8237 | . . . . 5 |
20 | 19 | con3d 620 | . . . 4 |
21 | 15, 10, 5, 16, 17, 18 | ltordlem 8237 | . . . . . 6 |
22 | 21 | con3d 620 | . . . . 5 |
23 | 22 | ancom2s 555 | . . . 4 |
24 | 20, 23 | anim12d 333 | . . 3 |
25 | 17 | ralrimiva 2503 | . . . . . 6 |
26 | 5 | eleq1d 2206 | . . . . . . 7 |
27 | 26 | rspccva 2783 | . . . . . 6 |
28 | 25, 27 | sylan 281 | . . . . 5 |
29 | 28 | adantrr 470 | . . . 4 |
30 | 10 | eleq1d 2206 | . . . . . . 7 |
31 | 30 | rspccva 2783 | . . . . . 6 |
32 | 25, 31 | sylan 281 | . . . . 5 |
33 | 32 | adantrl 469 | . . . 4 |
34 | 29, 33 | lttri3d 7871 | . . 3 |
35 | 16, 1 | sseldi 3090 | . . . 4 |
36 | simprr 521 | . . . . 5 | |
37 | 16, 36 | sseldi 3090 | . . . 4 |
38 | 35, 37 | lttri3d 7871 | . . 3 |
39 | 24, 34, 38 | 3imtr4d 202 | . 2 |
40 | 14, 39 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2414 wss 3066 class class class wbr 3924 cr 7612 clt 7793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-apti 7728 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: eqord2 8239 reef11 11395 |
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