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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 4358 | . . . . 5 | |
2 | ordsucss 4486 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | 3 | imp 123 | . . 3 |
5 | sssucid 4398 | . . 3 | |
6 | 4, 5 | sstrdi 3159 | . 2 |
7 | onelon 4367 | . . 3 | |
8 | elex 2741 | . . . 4 | |
9 | sucexg 4480 | . . . 4 | |
10 | suceq 4385 | . . . . 5 | |
11 | sucinc.1 | . . . . 5 | |
12 | 10, 11 | fvmptg 5570 | . . . 4 |
13 | 8, 9, 12 | syl2anc 409 | . . 3 |
14 | 7, 13 | syl 14 | . 2 |
15 | elex 2741 | . . . 4 | |
16 | sucexg 4480 | . . . 4 | |
17 | suceq 4385 | . . . . 5 | |
18 | 17, 11 | fvmptg 5570 | . . . 4 |
19 | 15, 16, 18 | syl2anc 409 | . . 3 |
20 | 19 | adantr 274 | . 2 |
21 | 6, 14, 20 | 3sstr4d 3192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cvv 2730 wss 3121 cmpt 4048 word 4345 con0 4346 csuc 4348 cfv 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 |
This theorem is referenced by: (None) |
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