Theorem nnpredcl 4745

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 4686) but also holds when it is (/) by uni0 3941. (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 3923	. . . 4 |- ( A = (/) -> U. A = U. (/) )
2		uni0 3941	. . . . 5 |- U. (/) = (/)
3		peano1 4716	. . . . 5 |- (/) e. om
4	2, 3	eqeltri 2305	. . . 4 |- U. (/) e. om
5	1, 4	eqeltrdi 2323	. . 3 |- ( A = (/) -> U. A e. om )
6	5	adantl 277	. 2 |- ( ( A e. om /\ A = (/) ) -> U. A e. om )
7		nnon 4732	. . . . . 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 4686	. . . . . . 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 2664	. . . . 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 2309	. . 3 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A e. om )
16		peano2b 4737	. . 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 4726	. 2 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
19	6, 17, 18	mpjaodan 806	1 |- ( A e. om -> U. A e. om )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   E.wrex 2521   (/)c0 3508   U.cuni 3914   Oncon0 4484   suc csuc 4486   omcom 4712

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713

This theorem is referenced by:  nnpredlt  4746  omp1eomlem  7385  ctmlemr  7399  nnnninfeq2  7420  nninfisollemne  7422  nninfisol  7424  nnsf  16783  peano4nninf  16784