ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uniintsnr GIF version

Theorem uniintsnr 3777
Description: The union and intersection of a singleton are equal. See also eusn 3567. (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 2663 . . . 4 𝑥 ∈ V
21unisn 3722 . . 3 {𝑥} = 𝑥
3 unieq 3715 . . 3 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 inteq 3744 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
51intsn 3776 . . . 4 {𝑥} = 𝑥
64, 5syl6eq 2166 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝑥)
72, 3, 63eqtr4a 2176 . 2 (𝐴 = {𝑥} → 𝐴 = 𝐴)
87exlimiv 1562 1 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wex 1453  {csn 3497   cuni 3706   cint 3741
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742
This theorem is referenced by:  uniintabim  3778
  Copyright terms: Public domain W3C validator