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Theorem uniprg 4928
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) Avoid using unipr 4929 to prove it from uniprg 4928. (Revised by BJ, 1-Sep-2024.)
Assertion
Ref Expression
uniprg ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Proof of Theorem uniprg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3482 . . . . . . . . 9 𝑦 ∈ V
21elpr 4655 . . . . . . . 8 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
32anbi2i 623 . . . . . . 7 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)))
4 ancom 460 . . . . . . . 8 ((𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) ∧ 𝑥𝑦))
5 andir 1010 . . . . . . . 8 (((𝑦 = 𝐴𝑦 = 𝐵) ∧ 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∨ (𝑦 = 𝐵𝑥𝑦)))
64, 5bitri 275 . . . . . . 7 ((𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴𝑥𝑦) ∨ (𝑦 = 𝐵𝑥𝑦)))
73, 6bitri 275 . . . . . 6 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑦 = 𝐴𝑥𝑦) ∨ (𝑦 = 𝐵𝑥𝑦)))
87exbii 1845 . . . . 5 (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑦 = 𝐴𝑥𝑦) ∨ (𝑦 = 𝐵𝑥𝑦)))
9 19.43 1880 . . . . 5 (∃𝑦((𝑦 = 𝐴𝑥𝑦) ∨ (𝑦 = 𝐵𝑥𝑦)) ↔ (∃𝑦(𝑦 = 𝐴𝑥𝑦) ∨ ∃𝑦(𝑦 = 𝐵𝑥𝑦)))
108, 9bitri 275 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (∃𝑦(𝑦 = 𝐴𝑥𝑦) ∨ ∃𝑦(𝑦 = 𝐵𝑥𝑦)))
11 clel3g 3661 . . . . . . 7 (𝐴𝑉 → (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦)))
1211bicomd 223 . . . . . 6 (𝐴𝑉 → (∃𝑦(𝑦 = 𝐴𝑥𝑦) ↔ 𝑥𝐴))
1312adantr 480 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑦(𝑦 = 𝐴𝑥𝑦) ↔ 𝑥𝐴))
14 clel3g 3661 . . . . . . 7 (𝐵𝑊 → (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦)))
1514bicomd 223 . . . . . 6 (𝐵𝑊 → (∃𝑦(𝑦 = 𝐵𝑥𝑦) ↔ 𝑥𝐵))
1615adantl 481 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑦(𝑦 = 𝐵𝑥𝑦) ↔ 𝑥𝐵))
1713, 16orbi12d 918 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∃𝑦(𝑦 = 𝐴𝑥𝑦) ∨ ∃𝑦(𝑦 = 𝐵𝑥𝑦)) ↔ (𝑥𝐴𝑥𝐵)))
1810, 17bitrid 283 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥𝐴𝑥𝐵)))
1918abbidv 2806 . 2 ((𝐴𝑉𝐵𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})} = {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
20 df-uni 4913 . 2 {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
21 df-un 3968 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
2219, 20, 213eqtr4g 2800 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  {cab 2712  cun 3961  {cpr 4633   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by:  unipr  4929  unisng  4930  unexg  7762  wunun  10748  tskun  10824  gruun  10844  mrcun  17667  unopn  22925  indistopon  23024  unconn  23453  limcun  25945  sshjval3  31383  prsiga  34112  unelsiga  34115  unelldsys  34139  measxun2  34191  measssd  34196  carsgsigalem  34297  carsgclctun  34303  pmeasmono  34306  probun  34401  indispconn  35219  bj-prmoore  37098  kelac2  43054  onsucunipr  43362  onsucunitp  43363  oaun2  43371  oaun3  43372  mnuund  44274  fourierdlem70  46132  fourierdlem71  46133  saluncl  46273  prsal  46274  meadjun  46418  omeunle  46472  toplatjoin  48791
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