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Theorem bj-nn0suc0 9410
Description: Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9424. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc0  om  (/)  suc
Distinct variable group:   ,

Proof of Theorem bj-nn0suc0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . 3  (/)  (/)
2 eqeq1 2043 . . . 4  suc  suc
32rexeqbi1dv 2508 . . 3  suc  suc
41, 3orbi12d 706 . 2  (/)  suc  (/)  suc
5 tru 1246 . . 3
6 a1tru 1258 . . . 4
76rgenw 2370 . . 3  om
8 bdeq0 9322 . . . . 5 BOUNDED  (/)
9 bdeqsuc 9336 . . . . . 6 BOUNDED  suc
109ax-bdex 9274 . . . . 5 BOUNDED  suc
118, 10ax-bdor 9271 . . . 4 BOUNDED  (/)  suc
12 nfv 1418 . . . 4  F/
13 orc 632 . . . . 5  (/)  (/)  suc
1413a1d 22 . . . 4  (/)  (/)  suc
15 a1tru 1258 . . . . 5  (/)  suc
1615expi 566 . . . 4  (/)  suc
17 vex 2554 . . . . . . . . 9 
_V
1817sucid 4120 . . . . . . . 8 
suc
19 eleq2 2098 . . . . . . . 8  suc  suc
2018, 19mpbiri 157 . . . . . . 7  suc
21 suceq 4105 . . . . . . . . 9  suc  suc
2221eqeq2d 2048 . . . . . . . 8  suc 
suc
2322rspcev 2650 . . . . . . 7  suc  suc
2420, 23mpancom 399 . . . . . 6  suc  suc
2524olcd 652 . . . . 5  suc  (/)  suc
2625a1d 22 . . . 4  suc  (/)  suc
2711, 12, 12, 12, 14, 16, 26bj-bdfindis 9407 . . 3  om  om  (/)  suc
285, 7, 27mp2an 402 . 2  om  (/)  suc
294, 28vtoclri 2622 1  om  (/)  suc
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wo 628   wceq 1242   wtru 1243   wcel 1390  wral 2300  wrex 2301   (/)c0 3218   suc csuc 4068   omcom 4256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdim 9269  ax-bdan 9270  ax-bdor 9271  ax-bdn 9272  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386
This theorem is referenced by:  bj-nn0suc  9424
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