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Theorem dmsnm 4713
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 4329 . 2 (A (V × V) ↔ xy A = ⟨x, y⟩)
2 vex 2538 . . . . 5 x V
32eldm 4459 . . . 4 (x dom {A} ↔ y x{A}y)
4 df-br 3739 . . . . . 6 (x{A}y ↔ ⟨x, y {A})
5 vex 2538 . . . . . . . 8 y V
62, 5opex 3940 . . . . . . 7 x, y V
76elsnc 3373 . . . . . 6 (⟨x, y {A} ↔ ⟨x, y⟩ = A)
8 eqcom 2024 . . . . . 6 (⟨x, y⟩ = AA = ⟨x, y⟩)
94, 7, 83bitri 195 . . . . 5 (x{A}yA = ⟨x, y⟩)
109exbii 1478 . . . 4 (y x{A}yy A = ⟨x, y⟩)
113, 10bitr2i 174 . . 3 (y A = ⟨x, y⟩ ↔ x dom {A})
1211exbii 1478 . 2 (xy A = ⟨x, y⟩ ↔ x x dom {A})
131, 12bitri 173 1 (A (V × V) ↔ x x dom {A})
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  {csn 3350  cop 3353   class class class wbr 3738   × cxp 4270  dom cdm 4272
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-dm 4282
This theorem is referenced by:  rnsnm  4714  dmsn0  4715  dmsn0el  4717  relsn2m  4718
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