Theorem nnpredcl 4607

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 4548) but also holds when it is (/) by uni0 3823. (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 3805	. . . 4 |- ( A = (/) -> U. A = U. (/) )
2		uni0 3823	. . . . 5 |- U. (/) = (/)
3		peano1 4578	. . . . 5 |- (/) e. om
4	2, 3	eqeltri 2243	. . . 4 |- U. (/) e. om
5	1, 4	eqeltrdi 2261	. . 3 |- ( A = (/) -> U. A e. om )
6	5	adantl 275	. 2 |- ( ( A e. om /\ A = (/) ) -> U. A e. om )
7		nnon 4594	. . . . . 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 4548	. . . . . . 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 2591	. . . . 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 2247	. . 3 |- ( ( A e. om /\ E. x e. om A = suc x ) -> suc U. A e. om )
16		peano2b 4599	. . 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 4588	. 2 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
19	6, 17, 18	mpjaodan 793	1 |- ( A e. om -> U. A e. om )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   (/)c0 3414   U.cuni 3796   Oncon0 4348   suc csuc 4350   omcom 4574

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575

This theorem is referenced by:  nnpredlt  4608  omp1eomlem  7071  ctmlemr  7085  nnnninfeq2  7105  nninfisollemne  7107  nninfisol  7109  nnsf  14038  peano4nninf  14039