Theorem nnsucpred 4759

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 4758	. 2 |- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
2		nnon 4752	. . . 4 |- ( A e. om -> A e. On )
3	2	ad2antrr 492	. . 3 |- ( ( ( A e. om /\ A =/= (/) ) /\ ( x e. om /\ A = suc x ) ) -> A e. On )
4		simprr 537	. . 3 |- ( ( ( A e. om /\ A =/= (/) ) /\ ( x e. om /\ A = suc x ) ) -> A = suc x )
5		onsucuni2 4706	. . 3 |- ( ( A e. On /\ A = suc x ) -> suc U. A = A )
6	3, 4, 5	syl2anc 415	. 2 |- ( ( ( A e. om /\ A =/= (/) ) /\ ( x e. om /\ A = suc x ) ) -> suc U. A = A )
7	1, 6	rexlimddv 2673	1 |- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   U.cuni 3930   Oncon0 4503   suc csuc 4505   omcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733

This theorem is referenced by:  nnpredlt  4766  omp1eomlem  7424  nnnninfeq2  7459  nnsf  16953