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Theorem el 3922
 Description: Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
el y x y
Distinct variable group:   x,y

Proof of Theorem el
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 zfpow 3919 . 2 yz(y(y zy x) → z y)
2 ax-14 1402 . . . . 5 (z = x → (y zy x))
32alrimiv 1751 . . . 4 (z = xy(y zy x))
4 ax-13 1401 . . . 4 (z = x → (z yx y))
53, 4embantd 50 . . 3 (z = x → ((y(y zy x) → z y) → x y))
65spimv 1689 . 2 (z(y(y zy x) → z y) → x y)
71, 6eximii 1490 1 y x y
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240  ∃wex 1378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-pow 3918 This theorem depends on definitions:  df-bi 110  df-nf 1347 This theorem is referenced by:  dtruarb  3933
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