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Theorem uniex2 7184
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 variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 zfun 7182 . . . 4 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 eluni 4629 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
32imbi1i 341 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
43albii 1915 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
54exbii 1944 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
61, 5mpbir 223 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
76bm1.3ii 4976 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
8 dfcleq 2791 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
98exbii 1944 . 2 (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦𝑧(𝑧𝑦𝑧 𝑥))
107, 9mpbir 223 1 𝑦 𝑦 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wex 1875  wcel 2157   cuni 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775  ax-sep 4973  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-uni 4627
This theorem is referenced by:  uniex  7185
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