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Theorem uniex2 7453
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 7451 . . . 4 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 eluni 4833 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
32imbi1i 351 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
43albii 1811 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
54exbii 1839 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
61, 5mpbir 232 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
76bm1.3ii 5197 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
8 dfcleq 2812 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
98exbii 1839 . 2 (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦𝑧(𝑧𝑦𝑧 𝑥))
107, 9mpbir 232 1 𝑦 𝑦 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-uni 4831
This theorem is referenced by:  uniex  7454
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