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Theorem uniun 3843
Description: The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
uniun (𝐴𝐵) = ( 𝐴 𝐵)

Proof of Theorem uniun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.43 1639 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elun 3291 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 457 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 andi 819 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
53, 4bitri 184 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
65exbii 1616 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
7 eluni 3827 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3827 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8orbi12i 765 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93bitr4i 212 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥 𝐴𝑥 𝐵))
11 eluni 3827 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elun 3291 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123bitr4i 212 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1413eqriv 2186 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wo 709   = wceq 1364  wex 1503  wcel 2160  cun 3142   cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-uni 3825
This theorem is referenced by:  unisuc  4431  unisucg  4432
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