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

Theorem dmsnm 4813
Description: The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
dmsnm (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem dmsnm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elvv 4429 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 2577 . . . . 5 𝑥 ∈ V
32eldm 4559 . . . 4 (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
4 df-br 3792 . . . . . 6 (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
5 vex 2577 . . . . . . . 8 𝑦 ∈ V
62, 5opex 3993 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
76elsn 3418 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
8 eqcom 2058 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴𝐴 = ⟨𝑥, 𝑦⟩)
94, 7, 83bitri 199 . . . . 5 (𝑥{𝐴}𝑦𝐴 = ⟨𝑥, 𝑦⟩)
109exbii 1512 . . . 4 (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
113, 10bitr2i 178 . . 3 (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
1211exbii 1512 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
131, 12bitri 177 1 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  {csn 3402  cop 3405   class class class wbr 3791   × cxp 4370  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-dm 4382
This theorem is referenced by:  rnsnm  4814  dmsn0  4815  dmsn0el  4817  relsn2m  4818
  Copyright terms: Public domain W3C validator