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Theorem uniex2 7454
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 7451 . . . 4 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
2 eluni 4827 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
32imbi1i 353 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦))
43albii 1821 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦))
54exbii 1849 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 234 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
76bm1.3ii 5192 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
8 dfcleq 2818 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
98exbii 1849 . 2 (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦𝑧(𝑧𝑦𝑧 𝑥))
107, 9mpbir 234 1 𝑦 𝑦 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115   cuni 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796  ax-sep 5189  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-uni 4825
This theorem is referenced by:  vuniex  7455
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