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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0sucALT | Unicode version |
Description: Alternate proof of bj-nn0suc 10759, also constructive but from ax-inf2 10771, hence requiring ax-bdsetind 10763. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-nn0sucALT |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-inf2 10771 | . . 3 | |
2 | vex 2604 | . . . . 5 | |
3 | bdcv 10639 | . . . . . 6 BOUNDED | |
4 | 3 | bj-inf2vn 10769 | . . . . 5 |
5 | 2, 4 | ax-mp 7 | . . . 4 |
6 | eleq2 2142 | . . . . . . 7 | |
7 | rexeq 2550 | . . . . . . . 8 | |
8 | 7 | orbi2d 736 | . . . . . . 7 |
9 | 6, 8 | bibi12d 233 | . . . . . 6 |
10 | 9 | albidv 1745 | . . . . 5 |
11 | nfcv 2219 | . . . . . . . 8 | |
12 | nfv 1461 | . . . . . . . 8 | |
13 | eleq1 2141 | . . . . . . . . . 10 | |
14 | eqeq1 2087 | . . . . . . . . . . 11 | |
15 | suceq 4157 | . . . . . . . . . . . . . 14 | |
16 | 15 | eqeq2d 2092 | . . . . . . . . . . . . 13 |
17 | 16 | cbvrexv 2578 | . . . . . . . . . . . 12 |
18 | eqeq1 2087 | . . . . . . . . . . . . 13 | |
19 | 18 | rexbidv 2369 | . . . . . . . . . . . 12 |
20 | 17, 19 | syl5bb 190 | . . . . . . . . . . 11 |
21 | 14, 20 | orbi12d 739 | . . . . . . . . . 10 |
22 | 13, 21 | bibi12d 233 | . . . . . . . . 9 |
23 | bi1 116 | . . . . . . . . 9 | |
24 | 22, 23 | syl6bi 161 | . . . . . . . 8 |
25 | 11, 12, 24 | spcimgf 2678 | . . . . . . 7 |
26 | 25 | pm2.43b 51 | . . . . . 6 |
27 | peano1 4335 | . . . . . . . 8 | |
28 | eleq1 2141 | . . . . . . . 8 | |
29 | 27, 28 | mpbiri 166 | . . . . . . 7 |
30 | bj-peano2 10734 | . . . . . . . . 9 | |
31 | eleq1a 2150 | . . . . . . . . . 10 | |
32 | 31 | imp 122 | . . . . . . . . 9 |
33 | 30, 32 | sylan 277 | . . . . . . . 8 |
34 | 33 | rexlimiva 2472 | . . . . . . 7 |
35 | 29, 34 | jaoi 668 | . . . . . 6 |
36 | 26, 35 | impbid1 140 | . . . . 5 |
37 | 10, 36 | syl6bi 161 | . . . 4 |
38 | 5, 37 | mpcom 36 | . . 3 |
39 | 1, 38 | eximii 1533 | . 2 |
40 | bj-ex 10573 | . 2 | |
41 | 39, 40 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wo 661 wal 1282 wceq 1284 wex 1421 wcel 1433 wrex 2349 cvv 2601 c0 3251 csuc 4120 com 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 ax-pr 3964 ax-un 4188 ax-bd0 10604 ax-bdim 10605 ax-bdor 10607 ax-bdex 10610 ax-bdeq 10611 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 ax-bdsetind 10763 ax-inf2 10771 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332 df-bdc 10632 df-bj-ind 10722 |
This theorem is referenced by: (None) |
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