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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0suc | Unicode version |
Description: Proof of (biconditional form of) nn0suc 4488 from the core axioms of CZF. See also bj-nn0sucALT 13103. As a characterization of the elements of , this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nn0suc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nn0suc0 13075 | . . 3 | |
2 | bj-omtrans 13081 | . . . . 5 | |
3 | ssrexv 3132 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | orim2d 762 | . . 3 |
6 | 1, 5 | mpd 13 | . 2 |
7 | peano1 4478 | . . . 4 | |
8 | eleq1 2180 | . . . 4 | |
9 | 7, 8 | mpbiri 167 | . . 3 |
10 | bj-peano2 13064 | . . . . 5 | |
11 | eleq1a 2189 | . . . . . 6 | |
12 | 11 | imp 123 | . . . . 5 |
13 | 10, 12 | sylan 281 | . . . 4 |
14 | 13 | rexlimiva 2521 | . . 3 |
15 | 9, 14 | jaoi 690 | . 2 |
16 | 6, 15 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 682 wceq 1316 wcel 1465 wrex 2394 wss 3041 c0 3333 csuc 4257 com 4474 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 ax-pr 4101 ax-un 4325 ax-bd0 12938 ax-bdim 12939 ax-bdan 12940 ax-bdor 12941 ax-bdn 12942 ax-bdal 12943 ax-bdex 12944 ax-bdeq 12945 ax-bdel 12946 ax-bdsb 12947 ax-bdsep 13009 ax-infvn 13066 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 df-bdc 12966 df-bj-ind 13052 |
This theorem is referenced by: bj-findis 13104 |
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