Theorem bj-nn0suc 13333

Description: Proof of (biconditional form of) nn0suc 4526 from the core axioms of CZF. See also bj-nn0sucALT 13347. As a characterization of the elements of om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nn0suc	|- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )

Distinct variable group:   x, A

Proof of Theorem bj-nn0suc

Step	Hyp	Ref	Expression
1		bj-nn0suc0 13319	. . 3 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
2		bj-omtrans 13325	. . . . 5 |- ( A e. om -> A C_ om )
3		ssrexv 3167	. . . . 5 |- ( A C_ om -> ( E. x e. A A = suc x -> E. x e. om A = suc x ) )
4	2, 3	syl 14	. . . 4 |- ( A e. om -> ( E. x e. A A = suc x -> E. x e. om A = suc x ) )
5	4	orim2d 778	. . 3 |- ( A e. om -> ( ( A = (/) \/ E. x e. A A = suc x ) -> ( A = (/) \/ E. x e. om A = suc x ) ) )
6	1, 5	mpd 13	. 2 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
7		peano1 4516	. . . 4 |- (/) e. om
8		eleq1 2203	. . . 4 |- ( A = (/) -> ( A e. om <-> (/) e. om ) )
9	7, 8	mpbiri 167	. . 3 |- ( A = (/) -> A e. om )
10		bj-peano2 13308	. . . . 5 |- ( x e. om -> suc x e. om )
11		eleq1a 2212	. . . . . 6 |- ( suc x e. om -> ( A = suc x -> A e. om ) )
12	11	imp 123	. . . . 5 |- ( ( suc x e. om /\ A = suc x ) -> A e. om )
13	10, 12	sylan 281	. . . 4 |- ( ( x e. om /\ A = suc x ) -> A e. om )
14	13	rexlimiva 2547	. . 3 |- ( E. x e. om A = suc x -> A e. om )
15	9, 14	jaoi 706	. 2 |- ( ( A = (/) \/ E. x e. om A = suc x ) -> A e. om )
16	6, 15	impbii 125	1 |- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   E.wrex 2418   C_ wss 3076   (/)c0 3368   suc csuc 4295   omcom 4512

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062  ax-pr 4139  ax-un 4363  ax-bd0 13182  ax-bdim 13183  ax-bdan 13184  ax-bdor 13185  ax-bdn 13186  ax-bdal 13187  ax-bdex 13188  ax-bdeq 13189  ax-bdel 13190  ax-bdsb 13191  ax-bdsep 13253  ax-infvn 13310

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-iom 4513  df-bdc 13210  df-bj-ind 13296

This theorem is referenced by:  bj-findis  13348