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Theorem uniintsnr 3692
Description: The union and intersection of a singleton are equal. See also eusn 3484. (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 2613 . . . 4 𝑥 ∈ V
21unisn 3637 . . 3 {𝑥} = 𝑥
3 unieq 3630 . . 3 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 inteq 3659 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
51intsn 3691 . . . 4 {𝑥} = 𝑥
64, 5syl6eq 2131 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝑥)
72, 3, 63eqtr4a 2141 . 2 (𝐴 = {𝑥} → 𝐴 = 𝐴)
87exlimiv 1530 1 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wex 1422  {csn 3416   cuni 3621   cint 3656
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657
This theorem is referenced by:  uniintabim  3693
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