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Theorem el 3956
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 𝑥𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem el
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 zfpow 3953 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
2 ax-14 1419 . . . . 5 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
32alrimiv 1768 . . . 4 (𝑧 = 𝑥 → ∀𝑦(𝑦𝑧𝑦𝑥))
4 ax-13 1418 . . . 4 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
53, 4embantd 54 . . 3 (𝑧 = 𝑥 → ((∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦))
65spimv 1706 . 2 (∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦)
71, 6eximii 1507 1 𝑦 𝑥𝑦
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1255  wex 1395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-pow 3952
This theorem depends on definitions:  df-bi 114  df-nf 1364
This theorem is referenced by:  dtruarb  3967
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