Theorem eqelsuc 4454

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 4452	. 2 |- A e. suc A
3		suceq 4437	. 2 |- ( A = B -> suc A = suc B )
4	2, 3	eleqtrid 2285	1 |- ( A = B -> A e. suc B )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   suc csuc 4400

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-suc 4406

This theorem is referenced by: (None)