Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sucinc2 | Unicode version |
Description: Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.) |
Ref | Expression |
---|---|
sucinc.1 |
Ref | Expression |
---|---|
sucinc2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4297 | . . . . 5 | |
2 | ordsucss 4420 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | 3 | imp 123 | . . 3 |
5 | sssucid 4337 | . . 3 | |
6 | 4, 5 | sstrdi 3109 | . 2 |
7 | onelon 4306 | . . 3 | |
8 | elex 2697 | . . . 4 | |
9 | sucexg 4414 | . . . 4 | |
10 | suceq 4324 | . . . . 5 | |
11 | sucinc.1 | . . . . 5 | |
12 | 10, 11 | fvmptg 5497 | . . . 4 |
13 | 8, 9, 12 | syl2anc 408 | . . 3 |
14 | 7, 13 | syl 14 | . 2 |
15 | elex 2697 | . . . 4 | |
16 | sucexg 4414 | . . . 4 | |
17 | suceq 4324 | . . . . 5 | |
18 | 17, 11 | fvmptg 5497 | . . . 4 |
19 | 15, 16, 18 | syl2anc 408 | . . 3 |
20 | 19 | adantr 274 | . 2 |
21 | 6, 14, 20 | 3sstr4d 3142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 wss 3071 cmpt 3989 word 4284 con0 4285 csuc 4287 cfv 5123 |
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-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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |