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Mirrors > Home > ILE Home > Th. List > phplem4dom | Unicode version |
Description: Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
phplem4dom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2 4318 | . . . . . 6 | |
2 | 1 | adantl 262 | . . . . 5 |
3 | brdomg 6229 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | biimpa 280 | . . 3 |
6 | simpr 103 | . . . . . . 7 | |
7 | 2 | ad2antrr 457 | . . . . . . 7 |
8 | sssucid 4152 | . . . . . . . 8 | |
9 | 8 | a1i 9 | . . . . . . 7 |
10 | simplll 485 | . . . . . . 7 | |
11 | f1imaen2g 6273 | . . . . . . 7 | |
12 | 6, 7, 9, 10, 11 | syl22anc 1136 | . . . . . 6 |
13 | 12 | ensymd 6263 | . . . . 5 |
14 | difexg 3898 | . . . . . . 7 | |
15 | 7, 14 | syl 14 | . . . . . 6 |
16 | nnord 4334 | . . . . . . . . . 10 | |
17 | orddif 4271 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | 18 | imaeq2d 4668 | . . . . . . . 8 |
20 | 10, 19 | syl 14 | . . . . . . 7 |
21 | f1fn 5093 | . . . . . . . . . . . 12 | |
22 | 21 | adantl 262 | . . . . . . . . . . 11 |
23 | sucidg 4153 | . . . . . . . . . . . 12 | |
24 | 10, 23 | syl 14 | . . . . . . . . . . 11 |
25 | fnsnfv 5232 | . . . . . . . . . . 11 | |
26 | 22, 24, 25 | syl2anc 391 | . . . . . . . . . 10 |
27 | 26 | difeq2d 3062 | . . . . . . . . 9 |
28 | df-f1 4907 | . . . . . . . . . . . 12 | |
29 | 28 | simprbi 260 | . . . . . . . . . . 11 |
30 | imadif 4979 | . . . . . . . . . . 11 | |
31 | 29, 30 | syl 14 | . . . . . . . . . 10 |
32 | 31 | adantl 262 | . . . . . . . . 9 |
33 | 27, 32 | eqtr4d 2075 | . . . . . . . 8 |
34 | f1f 5092 | . . . . . . . . . . 11 | |
35 | 34 | adantl 262 | . . . . . . . . . 10 |
36 | imassrn 4679 | . . . . . . . . . . 11 | |
37 | frn 5052 | . . . . . . . . . . 11 | |
38 | 36, 37 | syl5ss 2956 | . . . . . . . . . 10 |
39 | 35, 38 | syl 14 | . . . . . . . . 9 |
40 | 39 | ssdifd 3079 | . . . . . . . 8 |
41 | 33, 40 | eqsstr3d 2980 | . . . . . . 7 |
42 | 20, 41 | eqsstrd 2979 | . . . . . 6 |
43 | ssdomg 6258 | . . . . . 6 | |
44 | 15, 42, 43 | sylc 56 | . . . . 5 |
45 | endomtr 6270 | . . . . 5 | |
46 | 13, 44, 45 | syl2anc 391 | . . . 4 |
47 | simpllr 486 | . . . . . 6 | |
48 | 35, 24 | ffvelrnd 5303 | . . . . . 6 |
49 | phplem3g 6319 | . . . . . 6 | |
50 | 47, 48, 49 | syl2anc 391 | . . . . 5 |
51 | 50 | ensymd 6263 | . . . 4 |
52 | domentr 6271 | . . . 4 | |
53 | 46, 51, 52 | syl2anc 391 | . . 3 |
54 | 5, 53 | exlimddv 1778 | . 2 |
55 | 54 | ex 108 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cdif 2914 wss 2917 csn 3375 class class class wbr 3764 word 4099 csuc 4102 com 4313 ccnv 4344 crn 4346 cima 4348 wfun 4896 wfn 4897 wf 4898 wf1 4899 cfv 4902 cen 6219 cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 df-dom 6223 |
This theorem is referenced by: php5dom 6325 |
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