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

Theorem dmv 4579
Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
Assertion
Ref Expression
dmv dom V = V

Proof of Theorem dmv
StepHypRef Expression
1 ssv 2993 . 2 dom V ⊆ V
2 dmi 4578 . . 3 dom I = V
3 ssv 2993 . . . 4 I ⊆ V
4 dmss 4562 . . . 4 ( I ⊆ V → dom I ⊆ dom V)
53, 4ax-mp 7 . . 3 dom I ⊆ dom V
62, 5eqsstr3i 3004 . 2 V ⊆ dom V
71, 6eqssi 2989 1 dom V = V
Colors of variables: wff set class
Syntax hints:   = wceq 1259  Vcvv 2574  wss 2945   I cid 4053  dom cdm 4373
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 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-dm 4383
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator