Theorem eqelsuc 4522

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 4520	. 2 |- A e. suc A
3		suceq 4505	. 2 |- ( A = B -> suc A = suc B )
4	2, 3	eleqtrid 2320	1 |- ( A = B -> A e. suc B )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   suc csuc 4468

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-suc 4474

This theorem is referenced by: (None)