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Theorem el 4773
 Description: Every set is an element of some other set. See elALT 4837 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 4770 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
2 ax9 1990 . . . . 5 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
32alrimiv 1842 . . . 4 (𝑧 = 𝑥 → ∀𝑦(𝑦𝑧𝑦𝑥))
4 ax8 1983 . . . 4 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
53, 4embantd 57 . . 3 (𝑧 = 𝑥 → ((∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦))
65spimv 2245 . 2 (∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦)
71, 6eximii 1754 1 𝑦 𝑥𝑦
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  dtru  4783  dvdemo2  4830  axpownd  9302  zfcndinf  9319  domep  30942  distel  30953
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