Theorem nnpredcl 4623

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 4564) but also holds when it is (/) by uni0 3837. (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 3819	. . . 4 |- ( A = (/) -> U. A = U. (/) )
2		uni0 3837	. . . . 5 |- U. (/) = (/)
3		peano1 4594	. . . . 5 |- (/) e. om
4	2, 3	eqeltri 2250	. . . 4 |- U. (/) e. om
5	1, 4	eqeltrdi 2268	. . 3 |- ( A = (/) -> U. A e. om )
6	5	adantl 277	. 2 |- ( ( A e. om /\ A = (/) ) -> U. A e. om )
7		nnon 4610	. . . . . 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 4564	. . . . . . 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 2598	. . . . 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 2254	. . 3 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A e. om )
16		peano2b 4615	. . 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 4604	. 2 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
19	6, 17, 18	mpjaodan 798	1 |- ( A e. om -> U. A e. om )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   (/)c0 3423   U.cuni 3810   Oncon0 4364   suc csuc 4366   omcom 4590

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591

This theorem is referenced by:  nnpredlt  4624  omp1eomlem  7093  ctmlemr  7107  nnnninfeq2  7127  nninfisollemne  7129  nninfisol  7131  nnsf  14757  peano4nninf  14758