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Theorem iunxunsn 30625
Description: Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Hypothesis
Ref Expression
iunxunsn.1 (𝑥 = 𝑋𝐵 = 𝐶)
Assertion
Ref Expression
iunxunsn (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem iunxunsn
StepHypRef Expression
1 iunxun 5002 . 2 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋}𝐵)
2 iunxunsn.1 . . . 4 (𝑥 = 𝑋𝐵 = 𝐶)
32iunxsng 4998 . . 3 (𝑋𝑉 𝑥 ∈ {𝑋}𝐵 = 𝐶)
43uneq2d 4077 . 2 (𝑋𝑉 → ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋}𝐵) = ( 𝑥𝐴 𝐵𝐶))
51, 4syl5eq 2790 1 (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cun 3864  {csn 4541   ciun 4904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-un 3871  df-sn 4542  df-iun 4906
This theorem is referenced by: (None)
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