Theorem nnpredcl 4637

Description: The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4578) but also holds when it is (/) by uni0 3851. (Contributed by Jim Kingdon, 31-Jul-2022.)

Assertion

Ref	Expression
nnpredcl	|- ( A e. om -> U. A e. om )

Proof of Theorem nnpredcl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3833	. . . 4 |- ( A = (/) -> U. A = U. (/) )
2		uni0 3851	. . . . 5 |- U. (/) = (/)
3		peano1 4608	. . . . 5 |- (/) e. om
4	2, 3	eqeltri 2262	. . . 4 |- U. (/) e. om
5	1, 4	eqeltrdi 2280	. . 3 |- ( A = (/) -> U. A e. om )
6	5	adantl 277	. 2 |- ( ( A e. om /\ A = (/) ) -> U. A e. om )
7		nnon 4624	. . . . . 6 |- ( A e. om -> A e. On )
8	7	adantr 276	. . . . 5 |- ( ( A e. om /\ E. x e. om A = suc x ) -> A e. On )
9		simpr 110	. . . . 5 |- ( ( A e. om /\ E. x e. om A = suc x ) -> E. x e. om A = suc x )
10		onsucuni2 4578	. . . . . . 7 |- ( ( A e. On /\ A = suc x ) -> suc U. A = A )
11	10	ex 115	. . . . . 6 |- ( A e. On -> ( A = suc x -> suc U. A = A ) )
12	11	rexlimdvw 2611	. . . . 5 |- ( A e. On -> ( E. x e. om A = suc x -> suc U. A = A ) )
13	8, 9, 12	sylc 62	. . . 4 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A = A )
14		simpl 109	. . . 4 |- ( ( A e. om /\ E. x e. om A = suc x ) -> A e. om )
15	13, 14	eqeltrd 2266	. . 3 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A e. om )
16		peano2b 4629	. . 3 |- ( U. A e. om <-> suc U. A e. om )
17	15, 16	sylibr 134	. 2 |- ( ( A e. om /\ E. x e. om A = suc x ) -> U. A e. om )
18		nn0suc 4618	. 2 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
19	6, 17, 18	mpjaodan 799	1 |- ( A e. om -> U. A e. om )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   E.wrex 2469   (/)c0 3437   U.cuni 3824   Oncon0 4378   suc csuc 4380   omcom 4604

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605

This theorem is referenced by:  nnpredlt  4638  omp1eomlem  7111  ctmlemr  7125  nnnninfeq2  7145  nninfisollemne  7147  nninfisol  7149  nnsf  15139  peano4nninf  15140