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Theorem el 4838
Description: Every set is an element of some other set. See elALT 4901 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
el 𝑦 𝑥𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem el
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 zfpow 4835 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
2 ax9 2001 . . . . 5 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
32alrimiv 1853 . . . 4 (𝑧 = 𝑥 → ∀𝑦(𝑦𝑧𝑦𝑥))
4 ax8 1994 . . . 4 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
53, 4embantd 59 . . 3 (𝑧 = 𝑥 → ((∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦))
65spimv 2255 . 2 (∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦)
71, 6eximii 1762 1 𝑦 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-11 2032  ax-12 2045  ax-13 2244  ax-pow 4834
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  dtru  4848  dvdemo2  4894  axpownd  9408  zfcndinf  9425  domep  31672  distel  31683
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