Theorem nsuceq0g 4515

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 3498	. . 3 |- -. A e. (/)
2		sucidg 4513	. . . 4 |- ( A e. V -> A e. suc A )
3		eleq2 2295	. . . 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 670	. 2 |- ( A e. V -> -. suc A = (/) )
6	5	neneqad 2481	1 |- ( A e. V -> suc A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   =/= wne 2402   (/)c0 3494   suc csuc 4462

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-suc 4468

This theorem is referenced by:  onsucelsucexmid  4628  peano3  4694  frec0g  6562  2on0  6591  zfz1iso  11104