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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-peano4 | Unicode version |
Description: Remove from peano4 4506 dependency on ax-setind 4447. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-peano4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 978 | . . . . 5 | |
2 | pm3.22 263 | . . . . 5 | |
3 | bj-nnen2lp 13141 | . . . . 5 | |
4 | 1, 2, 3 | 3syl 17 | . . . 4 |
5 | sucidg 4333 | . . . . . . . . . . . 12 | |
6 | eleq2 2201 | . . . . . . . . . . . 12 | |
7 | 5, 6 | syl5ibrcom 156 | . . . . . . . . . . 11 |
8 | elsucg 4321 | . . . . . . . . . . 11 | |
9 | 7, 8 | sylibd 148 | . . . . . . . . . 10 |
10 | 9 | imp 123 | . . . . . . . . 9 |
11 | 10 | 3adant1 999 | . . . . . . . 8 |
12 | sucidg 4333 | . . . . . . . . . . . 12 | |
13 | eleq2 2201 | . . . . . . . . . . . 12 | |
14 | 12, 13 | syl5ibcom 154 | . . . . . . . . . . 11 |
15 | elsucg 4321 | . . . . . . . . . . 11 | |
16 | 14, 15 | sylibd 148 | . . . . . . . . . 10 |
17 | 16 | imp 123 | . . . . . . . . 9 |
18 | 17 | 3adant2 1000 | . . . . . . . 8 |
19 | 11, 18 | jca 304 | . . . . . . 7 |
20 | eqcom 2139 | . . . . . . . . 9 | |
21 | 20 | orbi2i 751 | . . . . . . . 8 |
22 | 21 | anbi1i 453 | . . . . . . 7 |
23 | 19, 22 | sylib 121 | . . . . . 6 |
24 | ordir 806 | . . . . . 6 | |
25 | 23, 24 | sylibr 133 | . . . . 5 |
26 | 25 | ord 713 | . . . 4 |
27 | 4, 26 | mpd 13 | . . 3 |
28 | 27 | 3expia 1183 | . 2 |
29 | suceq 4319 | . 2 | |
30 | 28, 29 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 csuc 4282 com 4499 |
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 603 ax-in2 604 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 2119 ax-nul 4049 ax-pr 4126 ax-un 4350 ax-bd0 13000 ax-bdor 13003 ax-bdn 13004 ax-bdal 13005 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 ax-infvn 13128 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: (None) |
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