Theorem nnsucpred 4616

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 4615	. 2 |- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
2		nnon 4609	. . . 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 4563	. . 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 2599	1 |- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   =/= wne 2347   (/)c0 3422   U.cuni 3809   Oncon0 4363   suc csuc 4365   omcom 4589

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590

This theorem is referenced by:  nnpredlt  4623  omp1eomlem  7092  nnnninfeq2  7126  nnsf  14636