Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > php5dom | Unicode version |
Description: A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
php5dom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceq 4374 | . . . 4 | |
2 | id 19 | . . . 4 | |
3 | 1, 2 | breq12d 3989 | . . 3 |
4 | 3 | notbid 657 | . 2 |
5 | suceq 4374 | . . . 4 | |
6 | id 19 | . . . 4 | |
7 | 5, 6 | breq12d 3989 | . . 3 |
8 | 7 | notbid 657 | . 2 |
9 | suceq 4374 | . . . 4 | |
10 | id 19 | . . . 4 | |
11 | 9, 10 | breq12d 3989 | . . 3 |
12 | 11 | notbid 657 | . 2 |
13 | suceq 4374 | . . . 4 | |
14 | id 19 | . . . 4 | |
15 | 13, 14 | breq12d 3989 | . . 3 |
16 | 15 | notbid 657 | . 2 |
17 | peano1 4565 | . . . 4 | |
18 | php5 6815 | . . . 4 | |
19 | 17, 18 | ax-mp 5 | . . 3 |
20 | 0ex 4103 | . . . . . 6 | |
21 | 20 | domen 6708 | . . . . 5 |
22 | ss0 3444 | . . . . . . . 8 | |
23 | en0 6752 | . . . . . . . 8 | |
24 | 22, 23 | sylibr 133 | . . . . . . 7 |
25 | entr 6741 | . . . . . . 7 | |
26 | 24, 25 | sylan2 284 | . . . . . 6 |
27 | 26 | exlimiv 1585 | . . . . 5 |
28 | 21, 27 | sylbi 120 | . . . 4 |
29 | 28 | ensymd 6740 | . . 3 |
30 | 19, 29 | mto 652 | . 2 |
31 | peano2 4566 | . . . 4 | |
32 | phplem4dom 6819 | . . . 4 | |
33 | 31, 32 | mpancom 419 | . . 3 |
34 | 33 | con3d 621 | . 2 |
35 | 4, 8, 12, 16, 30, 34 | finds 4571 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 wss 3111 c0 3404 class class class wbr 3976 csuc 4337 com 4561 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: nndomo 6821 phpm 6822 infnfi 6852 |
Copyright terms: Public domain | W3C validator |