MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  el Structured version   Visualization version   GIF version

Theorem el 4978
Description: Every set is an element of some other set. See elALT 5038 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 4975 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
2 ax9 2158 . . . . 5 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
32alrimiv 2007 . . . 4 (𝑧 = 𝑥 → ∀𝑦(𝑦𝑧𝑦𝑥))
4 ax8 2151 . . . 4 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
53, 4embantd 59 . . 3 (𝑧 = 𝑥 → ((∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦))
65spimv 2419 . 2 (∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦)
71, 6eximii 1912 1 𝑦 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-12 2203  ax-13 2408  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-nf 1858
This theorem is referenced by:  dtru  4988  dvdemo2  5031  axpownd  9625  zfcndinf  9642  domep  32034  distel  32045
  Copyright terms: Public domain W3C validator