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Theorem el 3960
 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 3957 . 2 𝑦𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦)
2 ax-14 1446 . . . . 5 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
32alrimiv 1796 . . . 4 (𝑧 = 𝑥 → ∀𝑦(𝑦𝑧𝑦𝑥))
4 ax-13 1445 . . . 4 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
53, 4embantd 55 . . 3 (𝑧 = 𝑥 → ((∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦))
65spimv 1733 . 2 (∀𝑧(∀𝑦(𝑦𝑧𝑦𝑥) → 𝑧𝑦) → 𝑥𝑦)
71, 6eximii 1534 1 𝑦 𝑥𝑦
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1283  ∃wex 1422 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-pow 3956 This theorem depends on definitions:  df-bi 115  df-nf 1391 This theorem is referenced by:  dtruarb  3970
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