Theorem nsuceq0g 4558

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 3525	. . 3 |- -. A e. (/)
2		sucidg 4556	. . . 4 |- ( A e. V -> A e. suc A )
3		eleq2 2302	. . . 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 674	. 2 |- ( A e. V -> -. suc A = (/) )
6	5	neneqad 2499	1 |- ( A e. V -> suc A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   suc csuc 4505

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-ext 2220

This theorem depends on definitions:  df-bi 117  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-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3711  df-suc 4511

This theorem is referenced by:  onsucelsucexmid  4672  peano3  4738  frec0g  6658  2on0  6687  zfz1iso  11271