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Theorem uniintsnr 3807
Description: The union and intersection of a singleton are equal. See also eusn 3597. (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 2689 . . . 4 𝑥 ∈ V
21unisn 3752 . . 3 {𝑥} = 𝑥
3 unieq 3745 . . 3 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 inteq 3774 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
51intsn 3806 . . . 4 {𝑥} = 𝑥
64, 5syl6eq 2188 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝑥)
72, 3, 63eqtr4a 2198 . 2 (𝐴 = {𝑥} → 𝐴 = 𝐴)
87exlimiv 1577 1 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wex 1468  {csn 3527   cuni 3736   cint 3771
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772
This theorem is referenced by:  uniintabim  3808
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