Theorem bj-nn0suc 15461

Description: Proof of (biconditional form of) nn0suc 4636 from the core axioms of CZF. See also bj-nn0sucALT 15475. 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 15447	. . 3 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
2		bj-omtrans 15453	. . . . 5 |- ( A e. om -> A C_ om )
3		ssrexv 3244	. . . . 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 789	. . 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 4626	. . . 4 |- (/) e. om
8		eleq1 2256	. . . 4 |- ( A = (/) -> ( A e. om <-> (/) e. om ) )
9	7, 8	mpbiri 168	. . 3 |- ( A = (/) -> A e. om )
10		bj-peano2 15436	. . . . 5 |- ( x e. om -> suc x e. om )
11		eleq1a 2265	. . . . . 6 |- ( suc x e. om -> ( A = suc x -> A e. om ) )
12	11	imp 124	. . . . 5 |- ( ( suc x e. om /\ A = suc x ) -> A e. om )
13	10, 12	sylan 283	. . . 4 |- ( ( x e. om /\ A = suc x ) -> A e. om )
14	13	rexlimiva 2606	. . 3 |- ( E. x e. om A = suc x -> A e. om )
15	9, 14	jaoi 717	. 2 |- ( ( A = (/) \/ E. x e. om A = suc x ) -> A e. om )
16	6, 15	impbii 126	1 |- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   E.wrex 2473   C_ wss 3153   (/)c0 3446   suc csuc 4396   omcom 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4464  ax-bd0 15310  ax-bdim 15311  ax-bdan 15312  ax-bdor 15313  ax-bdn 15314  ax-bdal 15315  ax-bdex 15316  ax-bdeq 15317  ax-bdel 15318  ax-bdsb 15319  ax-bdsep 15381  ax-infvn 15438

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623  df-bdc 15338  df-bj-ind 15424

This theorem is referenced by:  bj-findis  15476