Theorem eqelsuc 4510

Description: A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)

Hypothesis

Ref	Expression
eqelsuc.1	|- A e. _V

Assertion

Ref	Expression
eqelsuc	|- ( A = B -> A e. suc B )

Proof of Theorem eqelsuc

Step	Hyp	Ref	Expression
1		eqelsuc.1	. . 3 |- A e. _V
2	1	sucid 4508	. 2 |- A e. suc A
3		suceq 4493	. 2 |- ( A = B -> suc A = suc B )
4	2, 3	eleqtrid 2318	1 |- ( A = B -> A e. suc B )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   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-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-v 2801  df-un 3201  df-sn 3672  df-suc 4462

This theorem is referenced by: (None)