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Theorem uniintsnr 3707
Description: The union and intersection of a singleton are equal. See also eusn 3499. (Contributed by Jim Kingdon, 14-Aug-2018.)
Assertion
Ref Expression
uniintsnr (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem uniintsnr
StepHypRef Expression
1 vex 2618 . . . 4 𝑥 ∈ V
21unisn 3652 . . 3 {𝑥} = 𝑥
3 unieq 3645 . . 3 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 inteq 3674 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
51intsn 3706 . . . 4 {𝑥} = 𝑥
64, 5syl6eq 2133 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝑥)
72, 3, 63eqtr4a 2143 . 2 (𝐴 = {𝑥} → 𝐴 = 𝐴)
87exlimiv 1532 1 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wex 1424  {csn 3431   cuni 3636   cint 3671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-sn 3437  df-pr 3438  df-uni 3637  df-int 3672
This theorem is referenced by:  uniintabim  3708
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