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Theorem unisucs 6434
Description: The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6436. (Revised by BJ, 28-Dec-2024.)
Assertion
Ref Expression
unisucs (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))

Proof of Theorem unisucs
StepHypRef Expression
1 df-suc 6363 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
21unieqi 4914 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
32a1i 11 . 2 (𝐴𝑉 suc 𝐴 = (𝐴 ∪ {𝐴}))
4 uniun 4927 . . 3 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
54a1i 11 . 2 (𝐴𝑉 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴}))
6 unisng 4922 . . 3 (𝐴𝑉 {𝐴} = 𝐴)
76uneq2d 4158 . 2 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
83, 5, 73eqtrd 2770 1 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cun 3941  {csn 4623   cuni 4902  suc csuc 6359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-pr 4626  df-uni 4903  df-suc 6363
This theorem is referenced by:  unisucg  6435
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