Theorem bj-nn0suc 16099

Description: Proof of (biconditional form of) nn0suc 4670 from the core axioms of CZF. See also bj-nn0sucALT 16113. 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 16085	. . 3 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
2		bj-omtrans 16091	. . . . 5 |- ( A e. om -> A C_ om )
3		ssrexv 3266	. . . . 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 790	. . 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 4660	. . . 4 |- (/) e. om
8		eleq1 2270	. . . 4 |- ( A = (/) -> ( A e. om <-> (/) e. om ) )
9	7, 8	mpbiri 168	. . 3 |- ( A = (/) -> A e. om )
10		bj-peano2 16074	. . . . 5 |- ( x e. om -> suc x e. om )
11		eleq1a 2279	. . . . . 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 2620	. . 3 |- ( E. x e. om A = suc x -> A e. om )
15	9, 14	jaoi 718	. 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 710   = wceq 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   (/)c0 3468   suc csuc 4430   omcom 4656

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-nul 4186  ax-pr 4269  ax-un 4498  ax-bd0 15948  ax-bdim 15949  ax-bdan 15950  ax-bdor 15951  ax-bdn 15952  ax-bdal 15953  ax-bdex 15954  ax-bdeq 15955  ax-bdel 15956  ax-bdsb 15957  ax-bdsep 16019  ax-infvn 16076

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657  df-bdc 15976  df-bj-ind 16062

This theorem is referenced by:  bj-findis  16114