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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 4588 | . . . . . 6 | |
2 | 1 | adantl 277 | . . . . 5 |
3 | brdomg 6738 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | biimpa 296 | . . 3 |
6 | simpr 110 | . . . . . . 7 | |
7 | 2 | ad2antrr 488 | . . . . . . 7 |
8 | sssucid 4409 | . . . . . . . 8 | |
9 | 8 | a1i 9 | . . . . . . 7 |
10 | simplll 533 | . . . . . . 7 | |
11 | f1imaen2g 6783 | . . . . . . 7 | |
12 | 6, 7, 9, 10, 11 | syl22anc 1239 | . . . . . 6 |
13 | 12 | ensymd 6773 | . . . . 5 |
14 | difexg 4139 | . . . . . . 7 | |
15 | 7, 14 | syl 14 | . . . . . 6 |
16 | nnord 4605 | . . . . . . . . . 10 | |
17 | orddif 4540 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | 18 | imaeq2d 4963 | . . . . . . . 8 |
20 | 10, 19 | syl 14 | . . . . . . 7 |
21 | f1fn 5415 | . . . . . . . . . . . 12 | |
22 | 21 | adantl 277 | . . . . . . . . . . 11 |
23 | sucidg 4410 | . . . . . . . . . . . 12 | |
24 | 10, 23 | syl 14 | . . . . . . . . . . 11 |
25 | fnsnfv 5567 | . . . . . . . . . . 11 | |
26 | 22, 24, 25 | syl2anc 411 | . . . . . . . . . 10 |
27 | 26 | difeq2d 3251 | . . . . . . . . 9 |
28 | df-f1 5213 | . . . . . . . . . . . 12 | |
29 | 28 | simprbi 275 | . . . . . . . . . . 11 |
30 | imadif 5288 | . . . . . . . . . . 11 | |
31 | 29, 30 | syl 14 | . . . . . . . . . 10 |
32 | 31 | adantl 277 | . . . . . . . . 9 |
33 | 27, 32 | eqtr4d 2211 | . . . . . . . 8 |
34 | f1f 5413 | . . . . . . . . . . 11 | |
35 | 34 | adantl 277 | . . . . . . . . . 10 |
36 | imassrn 4974 | . . . . . . . . . . 11 | |
37 | frn 5366 | . . . . . . . . . . 11 | |
38 | 36, 37 | sstrid 3164 | . . . . . . . . . 10 |
39 | 35, 38 | syl 14 | . . . . . . . . 9 |
40 | 39 | ssdifd 3269 | . . . . . . . 8 |
41 | 33, 40 | eqsstrrd 3190 | . . . . . . 7 |
42 | 20, 41 | eqsstrd 3189 | . . . . . 6 |
43 | ssdomg 6768 | . . . . . 6 | |
44 | 15, 42, 43 | sylc 62 | . . . . 5 |
45 | endomtr 6780 | . . . . 5 | |
46 | 13, 44, 45 | syl2anc 411 | . . . 4 |
47 | simpllr 534 | . . . . . 6 | |
48 | 35, 24 | ffvelcdmd 5644 | . . . . . 6 |
49 | phplem3g 6846 | . . . . . 6 | |
50 | 47, 48, 49 | syl2anc 411 | . . . . 5 |
51 | 50 | ensymd 6773 | . . . 4 |
52 | domentr 6781 | . . . 4 | |
53 | 46, 51, 52 | syl2anc 411 | . . 3 |
54 | 5, 53 | exlimddv 1896 | . 2 |
55 | 54 | ex 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wex 1490 wcel 2146 cvv 2735 cdif 3124 wss 3127 csn 3589 class class class wbr 3998 word 4356 csuc 4359 com 4583 ccnv 4619 crn 4621 cima 4623 wfun 5202 wfn 5203 wf 5204 wf1 5205 cfv 5208 cen 6728 cdom 6729 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-er 6525 df-en 6731 df-dom 6732 |
This theorem is referenced by: php5dom 6853 |
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