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Theorem uniex2 7758
Description: The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
uniex2 𝑦 𝑦 = 𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem uniex2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-un 7755 . . . 4 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
2 eluni 4910 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
32imbi1i 349 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦))
43albii 1819 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦))
54exbii 1848 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 231 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
76sepexi 5301 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
8 dfcleq 2730 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
98exbii 1848 . 2 (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦𝑧(𝑧𝑦𝑧 𝑥))
107, 9mpbir 231 1 𝑦 𝑦 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-uni 4908
This theorem is referenced by:  vuniex  7759
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