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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0suc0 | Unicode version |
Description: Constructive proof of a variant of nn0suc 4488. For a constructive proof of nn0suc 4488, see bj-nn0suc 13089. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nn0suc0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2124 | . . 3 | |
2 | eqeq1 2124 | . . . 4 | |
3 | 2 | rexeqbi1dv 2612 | . . 3 |
4 | 1, 3 | orbi12d 767 | . 2 |
5 | tru 1320 | . . 3 | |
6 | a1tru 1332 | . . . 4 | |
7 | 6 | rgenw 2464 | . . 3 |
8 | bdeq0 12992 | . . . . 5 BOUNDED | |
9 | bdeqsuc 13006 | . . . . . 6 BOUNDED | |
10 | 9 | ax-bdex 12944 | . . . . 5 BOUNDED |
11 | 8, 10 | ax-bdor 12941 | . . . 4 BOUNDED |
12 | nfv 1493 | . . . 4 | |
13 | orc 686 | . . . . 5 | |
14 | 13 | a1d 22 | . . . 4 |
15 | a1tru 1332 | . . . . 5 | |
16 | 15 | expi 612 | . . . 4 |
17 | vex 2663 | . . . . . . . . 9 | |
18 | 17 | sucid 4309 | . . . . . . . 8 |
19 | eleq2 2181 | . . . . . . . 8 | |
20 | 18, 19 | mpbiri 167 | . . . . . . 7 |
21 | suceq 4294 | . . . . . . . . 9 | |
22 | 21 | eqeq2d 2129 | . . . . . . . 8 |
23 | 22 | rspcev 2763 | . . . . . . 7 |
24 | 20, 23 | mpancom 418 | . . . . . 6 |
25 | 24 | olcd 708 | . . . . 5 |
26 | 25 | a1d 22 | . . . 4 |
27 | 11, 12, 12, 12, 14, 16, 26 | bj-bdfindis 13072 | . . 3 |
28 | 5, 7, 27 | mp2an 422 | . 2 |
29 | 4, 28 | vtoclri 2735 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 682 wceq 1316 wtru 1317 wcel 1465 wral 2393 wrex 2394 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-nn0suc 13089 |
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