Theorem nnsucpred 4666

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 4665	. 2 |- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
2		nnon 4659	. . . 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 4613	. . 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 2628	1 |- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   =/= wne 2376   (/)c0 3460   U.cuni 3850   Oncon0 4411   suc csuc 4413   omcom 4639

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4144  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640

This theorem is referenced by:  nnpredlt  4673  omp1eomlem  7198  nnnninfeq2  7233  nnsf  15979