Theorem nnsucpred 4649

Description: The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)

Assertion

Ref	Expression
nnsucpred	|- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )

Proof of Theorem nnsucpred

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnsuc 4648	. 2 |- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
2		nnon 4642	. . . 4 |- ( A e. om -> A e. On )
3	2	ad2antrr 488	. . 3 |- ( ( ( A e. om /\ A =/= (/) ) /\ ( x e. om /\ A = suc x ) ) -> A e. On )
4		simprr 531	. . 3 |- ( ( ( A e. om /\ A =/= (/) ) /\ ( x e. om /\ A = suc x ) ) -> A = suc x )
5		onsucuni2 4596	. . 3 |- ( ( A e. On /\ A = suc x ) -> suc U. A = A )
6	3, 4, 5	syl2anc 411	. 2 |- ( ( ( A e. om /\ A =/= (/) ) /\ ( x e. om /\ A = suc x ) ) -> suc U. A = A )
7	1, 6	rexlimddv 2616	1 |- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364   (/)c0 3446   U.cuni 3835   Oncon0 4394   suc csuc 4396   omcom 4622

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623

This theorem is referenced by:  nnpredlt  4656  omp1eomlem  7153  nnnninfeq2  7188  nnsf  15495