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Theorem uniintsnr 3651
 Description: The union and intersection of a singleton are equal. See also eusn 3444. (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 2560 . . . 4 𝑥 ∈ V
21unisn 3596 . . 3 {𝑥} = 𝑥
3 unieq 3589 . . 3 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 inteq 3618 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
51intsn 3650 . . . 4 {𝑥} = 𝑥
64, 5syl6eq 2088 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝑥)
72, 3, 63eqtr4a 2098 . 2 (𝐴 = {𝑥} → 𝐴 = 𝐴)
87exlimiv 1489 1 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ∃wex 1381  {csn 3375  ∪ cuni 3580  ∩ cint 3615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616 This theorem is referenced by:  uniintabim  3652
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