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Theorem el 5399
Description: Any set is an element of some other set. See elALT 5402 for a shorter proof using more axioms, and see elALT2 5329 for a proof that uses ax-9 2117 and ax-pow 5325 instead of ax-pr 5389. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5389 instead of ax-9 2117 and ax-pow 5325. (Revised by BTernaryTau, 2-Dec-2024.)
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 ax-pr 5389 . 2 𝑦𝑧((𝑧 = 𝑥𝑧 = 𝑥) → 𝑧𝑦)
2 pm4.25 905 . . . . . 6 (𝑧 = 𝑥 ↔ (𝑧 = 𝑥𝑧 = 𝑥))
32imbi1i 350 . . . . 5 ((𝑧 = 𝑥𝑧𝑦) ↔ ((𝑧 = 𝑥𝑧 = 𝑥) → 𝑧𝑦))
43albii 1822 . . . 4 (∀𝑧(𝑧 = 𝑥𝑧𝑦) ↔ ∀𝑧((𝑧 = 𝑥𝑧 = 𝑥) → 𝑧𝑦))
5 elequ1 2114 . . . . 5 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
65equsalvw 2008 . . . 4 (∀𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
74, 6bitr3i 277 . . 3 (∀𝑧((𝑧 = 𝑥𝑧 = 𝑥) → 𝑧𝑦) ↔ 𝑥𝑦)
87exbii 1851 . 2 (∃𝑦𝑧((𝑧 = 𝑥𝑧 = 𝑥) → 𝑧𝑦) ↔ ∃𝑦 𝑥𝑦)
91, 8mpbi 229 1 𝑦 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783
This theorem is referenced by:  sels  5400  dtruOLD  5403  dmep  5884  axpownd  10544  zfcndinf  10561  distel  34417
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