Theorem eqelsuc 4413

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 4411	. 2 |- A e. suc A
3		suceq 4396	. 2 |- ( A = B -> suc A = suc B )
4	2, 3	eleqtrid 2264	1 |- ( A = B -> A e. suc B )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   _Vcvv 2735   suc csuc 4359

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-suc 4365

This theorem is referenced by: (None)