Theorem bj-nn0suc 15900

Description: Proof of (biconditional form of) nn0suc 4652 from the core axioms of CZF. See also bj-nn0sucALT 15914. 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 15886	. . 3 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
2		bj-omtrans 15892	. . . . 5 |- ( A e. om -> A C_ om )
3		ssrexv 3258	. . . . 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 4642	. . . 4 |- (/) e. om
8		eleq1 2268	. . . 4 |- ( A = (/) -> ( A e. om <-> (/) e. om ) )
9	7, 8	mpbiri 168	. . 3 |- ( A = (/) -> A e. om )
10		bj-peano2 15875	. . . . 5 |- ( x e. om -> suc x e. om )
11		eleq1a 2277	. . . . . 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 2618	. . 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 2176   E.wrex 2485   C_ wss 3166   (/)c0 3460   suc csuc 4412   omcom 4638

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-bd0 15749  ax-bdim 15750  ax-bdan 15751  ax-bdor 15752  ax-bdn 15753  ax-bdal 15754  ax-bdex 15755  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758  ax-bdsep 15820  ax-infvn 15877

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15777  df-bj-ind 15863

This theorem is referenced by:  bj-findis  15915