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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nn0suc0 | Unicode version |
Description: Constructive proof of a variant of nn0suc 4578. For a constructive proof of nn0suc 4578, see bj-nn0suc 13739. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nn0suc0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2171 | . . 3 | |
2 | eqeq1 2171 | . . . 4 | |
3 | 2 | rexeqbi1dv 2668 | . . 3 |
4 | 1, 3 | orbi12d 783 | . 2 |
5 | tru 1346 | . . 3 | |
6 | a1tru 1358 | . . . 4 | |
7 | 6 | rgenw 2519 | . . 3 |
8 | bdeq0 13642 | . . . . 5 BOUNDED | |
9 | bdeqsuc 13656 | . . . . . 6 BOUNDED | |
10 | 9 | ax-bdex 13594 | . . . . 5 BOUNDED |
11 | 8, 10 | ax-bdor 13591 | . . . 4 BOUNDED |
12 | nfv 1515 | . . . 4 | |
13 | orc 702 | . . . . 5 | |
14 | 13 | a1d 22 | . . . 4 |
15 | a1tru 1358 | . . . . 5 | |
16 | 15 | expi 628 | . . . 4 |
17 | vex 2727 | . . . . . . . . 9 | |
18 | 17 | sucid 4392 | . . . . . . . 8 |
19 | eleq2 2228 | . . . . . . . 8 | |
20 | 18, 19 | mpbiri 167 | . . . . . . 7 |
21 | suceq 4377 | . . . . . . . . 9 | |
22 | 21 | eqeq2d 2176 | . . . . . . . 8 |
23 | 22 | rspcev 2828 | . . . . . . 7 |
24 | 20, 23 | mpancom 419 | . . . . . 6 |
25 | 24 | olcd 724 | . . . . 5 |
26 | 25 | a1d 22 | . . . 4 |
27 | 11, 12, 12, 12, 14, 16, 26 | bj-bdfindis 13722 | . . 3 |
28 | 5, 7, 27 | mp2an 423 | . 2 |
29 | 4, 28 | vtoclri 2799 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 698 wceq 1342 wtru 1343 wcel 2135 wral 2442 wrex 2443 c0 3407 csuc 4340 com 4564 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4105 ax-pr 4184 ax-un 4408 ax-bd0 13588 ax-bdim 13589 ax-bdan 13590 ax-bdor 13591 ax-bdn 13592 ax-bdal 13593 ax-bdex 13594 ax-bdeq 13595 ax-bdel 13596 ax-bdsb 13597 ax-bdsep 13659 ax-infvn 13716 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2726 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-sn 3579 df-pr 3580 df-uni 3787 df-int 3822 df-suc 4346 df-iom 4565 df-bdc 13616 df-bj-ind 13702 |
This theorem is referenced by: bj-nn0suc 13739 |
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