Theorem nnpredcl 4582

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 4523) but also holds when it is (/) by uni0 3799. (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 3781	. . . 4 |- ( A = (/) -> U. A = U. (/) )
2		uni0 3799	. . . . 5 |- U. (/) = (/)
3		peano1 4553	. . . . 5 |- (/) e. om
4	2, 3	eqeltri 2230	. . . 4 |- U. (/) e. om
5	1, 4	eqeltrdi 2248	. . 3 |- ( A = (/) -> U. A e. om )
6	5	adantl 275	. 2 |- ( ( A e. om /\ A = (/) ) -> U. A e. om )
7		nnon 4569	. . . . . 6 |- ( A e. om -> A e. On )
8	7	adantr 274	. . . . 5 |- ( ( A e. om /\ E. x e. om A = suc x ) -> A e. On )
9		simpr 109	. . . . 5 |- ( ( A e. om /\ E. x e. om A = suc x ) -> E. x e. om A = suc x )
10		onsucuni2 4523	. . . . . . 7 |- ( ( A e. On /\ A = suc x ) -> suc U. A = A )
11	10	ex 114	. . . . . 6 |- ( A e. On -> ( A = suc x -> suc U. A = A ) )
12	11	rexlimdvw 2578	. . . . 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 108	. . . 4 |- ( ( A e. om /\ E. x e. om A = suc x ) -> A e. om )
15	13, 14	eqeltrd 2234	. . 3 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A e. om )
16		peano2b 4574	. . 3 |- ( U. A e. om <-> suc U. A e. om )
17	15, 16	sylibr 133	. 2 |- ( ( A e. om /\ E. x e. om A = suc x ) -> U. A e. om )
18		nn0suc 4563	. 2 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
19	6, 17, 18	mpjaodan 788	1 |- ( A e. om -> U. A e. om )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   E.wrex 2436   (/)c0 3394   U.cuni 3772   Oncon0 4323   suc csuc 4325   omcom 4549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550

This theorem is referenced by:  nnpredlt  4583  omp1eomlem  7038  ctmlemr  7052  nnnninfeq2  7072  nninfisollemne  7074  nninfisol  7076  nnsf  13577  peano4nninf  13578