Theorem nsuceq0g 4509

Description: No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)

Assertion

Ref	Expression
nsuceq0g	|- ( A e. V -> suc A =/= (/) )

Proof of Theorem nsuceq0g

Step	Hyp	Ref	Expression
1		noel 3495	. . 3 |- -. A e. (/)
2		sucidg 4507	. . . 4 |- ( A e. V -> A e. suc A )
3		eleq2 2293	. . . 4 |- ( suc A = (/) -> ( A e. suc A <-> A e. (/) ) )
4	2, 3	syl5ibcom 155	. . 3 |- ( A e. V -> ( suc A = (/) -> A e. (/) ) )
5	1, 4	mtoi 668	. 2 |- ( A e. V -> -. suc A = (/) )
6	5	neneqad 2479	1 |- ( A e. V -> suc A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   suc csuc 4456

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-suc 4462

This theorem is referenced by:  onsucelsucexmid  4622  peano3  4688  frec0g  6543  2on0  6572  zfz1iso  11063