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Theorem uniun 3723
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 1590 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elun 3185 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 450 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 andi 790 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
53, 4bitri 183 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
65exbii 1567 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
7 eluni 3707 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3707 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8orbi12i 736 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93bitr4i 211 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥 𝐴𝑥 𝐵))
11 eluni 3707 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elun 3185 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123bitr4i 211 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1413eqriv 2112 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wo 680   = wceq 1314  wex 1451  wcel 1463  cun 3037   cuni 3704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-uni 3705
This theorem is referenced by:  unisuc  4303  unisucg  4304
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