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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 4566 | . . . . . 6 | |
2 | 1 | adantl 275 | . . . . 5 |
3 | brdomg 6705 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | biimpa 294 | . . 3 |
6 | simpr 109 | . . . . . . 7 | |
7 | 2 | ad2antrr 480 | . . . . . . 7 |
8 | sssucid 4387 | . . . . . . . 8 | |
9 | 8 | a1i 9 | . . . . . . 7 |
10 | simplll 523 | . . . . . . 7 | |
11 | f1imaen2g 6750 | . . . . . . 7 | |
12 | 6, 7, 9, 10, 11 | syl22anc 1228 | . . . . . 6 |
13 | 12 | ensymd 6740 | . . . . 5 |
14 | difexg 4117 | . . . . . . 7 | |
15 | 7, 14 | syl 14 | . . . . . 6 |
16 | nnord 4583 | . . . . . . . . . 10 | |
17 | orddif 4518 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | 18 | imaeq2d 4940 | . . . . . . . 8 |
20 | 10, 19 | syl 14 | . . . . . . 7 |
21 | f1fn 5389 | . . . . . . . . . . . 12 | |
22 | 21 | adantl 275 | . . . . . . . . . . 11 |
23 | sucidg 4388 | . . . . . . . . . . . 12 | |
24 | 10, 23 | syl 14 | . . . . . . . . . . 11 |
25 | fnsnfv 5539 | . . . . . . . . . . 11 | |
26 | 22, 24, 25 | syl2anc 409 | . . . . . . . . . 10 |
27 | 26 | difeq2d 3235 | . . . . . . . . 9 |
28 | df-f1 5187 | . . . . . . . . . . . 12 | |
29 | 28 | simprbi 273 | . . . . . . . . . . 11 |
30 | imadif 5262 | . . . . . . . . . . 11 | |
31 | 29, 30 | syl 14 | . . . . . . . . . 10 |
32 | 31 | adantl 275 | . . . . . . . . 9 |
33 | 27, 32 | eqtr4d 2200 | . . . . . . . 8 |
34 | f1f 5387 | . . . . . . . . . . 11 | |
35 | 34 | adantl 275 | . . . . . . . . . 10 |
36 | imassrn 4951 | . . . . . . . . . . 11 | |
37 | frn 5340 | . . . . . . . . . . 11 | |
38 | 36, 37 | sstrid 3148 | . . . . . . . . . 10 |
39 | 35, 38 | syl 14 | . . . . . . . . 9 |
40 | 39 | ssdifd 3253 | . . . . . . . 8 |
41 | 33, 40 | eqsstrrd 3174 | . . . . . . 7 |
42 | 20, 41 | eqsstrd 3173 | . . . . . 6 |
43 | ssdomg 6735 | . . . . . 6 | |
44 | 15, 42, 43 | sylc 62 | . . . . 5 |
45 | endomtr 6747 | . . . . 5 | |
46 | 13, 44, 45 | syl2anc 409 | . . . 4 |
47 | simpllr 524 | . . . . . 6 | |
48 | 35, 24 | ffvelrnd 5615 | . . . . . 6 |
49 | phplem3g 6813 | . . . . . 6 | |
50 | 47, 48, 49 | syl2anc 409 | . . . . 5 |
51 | 50 | ensymd 6740 | . . . 4 |
52 | domentr 6748 | . . . 4 | |
53 | 46, 51, 52 | syl2anc 409 | . . 3 |
54 | 5, 53 | exlimddv 1885 | . 2 |
55 | 54 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2721 cdif 3108 wss 3111 csn 3570 class class class wbr 3976 word 4334 csuc 4337 com 4561 ccnv 4597 crn 4599 cima 4601 wfun 5176 wfn 5177 wf 5178 wf1 5179 cfv 5182 cen 6695 cdom 6696 |
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-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 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-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-er 6492 df-en 6698 df-dom 6699 |
This theorem is referenced by: php5dom 6820 |
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