Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  unisn3 GIF version

Theorem unisn3 4180
 Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
Assertion
Ref Expression
unisn3 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unisn3
StepHypRef Expression
1 rabsn 3437 . . 3 (𝐴𝐵 → {𝑥𝐵𝑥 = 𝐴} = {𝐴})
21unieqd 3591 . 2 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = {𝐴})
3 unisng 3597 . 2 (𝐴𝐵 {𝐴} = 𝐴)
42, 3eqtrd 2072 1 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  {crab 2310  {csn 3375  ∪ cuni 3580 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-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator