Theorem elrel 3249

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 3181	. . . . 5 |- (Rel R <-> R (_ (V X. V))
2	1	biimp 151	. . . 4 |- (Rel R -> R (_ (V X. V))
3	2	sseld 2064	. . 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 3224	. 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 955   e. wcel 957  E.wex 979  Vcvv 1808   (_ wss 2044  <.cop 2408   X. cxp 3164  Rel wrel 3171

This theorem is referenced by:  unielrel 3510

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-opab 2663  df-xp 3180  df-rel 3181

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