Theorem elrel 3253

Description: A member of a relation is an ordered pair.

Assertion

Ref	Expression
elrel	|- ((Rel R /\ A e. R) -> E.xE.y A = <.x, y>.)

Distinct variable group:   x,y,A

Proof of Theorem elrel

Step	Hyp	Ref	Expression
1		df-rel 3185	. . . . 5 |- (Rel R <-> R (_ (V X. V))
2	1	biimp 151	. . . 4 |- (Rel R -> R (_ (V X. V))
3	2	sseld 2067	. . 3 |- (Rel R -> (A e. R -> A e. (V X. V)))
4	3	imp 350	. 2 |- ((Rel R /\ A e. R) -> A e. (V X. V))
5		elvv 3228	. 2 |- (A e. (V X. V) <-> E.xE.y A = <.x, y>.)
6	4, 5	sylib 198	1 |- ((Rel R /\ A e. R) -> E.xE.y A = <.x, y>.)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811   (_ wss 2047  <.cop 2411   X. cxp 3168  Rel wrel 3175

This theorem is referenced by:  unielrel 3514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185

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