Theorem nnpredcl 4660

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 4601) but also holds when it is (/) by uni0 3867. (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 3849	. . . 4 |- ( A = (/) -> U. A = U. (/) )
2		uni0 3867	. . . . 5 |- U. (/) = (/)
3		peano1 4631	. . . . 5 |- (/) e. om
4	2, 3	eqeltri 2269	. . . 4 |- U. (/) e. om
5	1, 4	eqeltrdi 2287	. . 3 |- ( A = (/) -> U. A e. om )
6	5	adantl 277	. 2 |- ( ( A e. om /\ A = (/) ) -> U. A e. om )
7		nnon 4647	. . . . . 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 4601	. . . . . . 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 2618	. . . . 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 2273	. . 3 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A e. om )
16		peano2b 4652	. . 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 4641	. 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 2167   E.wrex 2476   (/)c0 3451   U.cuni 3840   Oncon0 4399   suc csuc 4401   omcom 4627

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628

This theorem is referenced by:  nnpredlt  4661  omp1eomlem  7169  ctmlemr  7183  nnnninfeq2  7204  nninfisollemne  7206  nninfisol  7208  nnsf  15736  peano4nninf  15737