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

Theorem vss 3470
Description: Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
vss (V ⊆ 𝐴𝐴 = V)

Proof of Theorem vss
StepHypRef Expression
1 ssv 3177 . . 3 𝐴 ⊆ V
21biantrur 303 . 2 (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
3 eqss 3170 . 2 (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
42, 3bitr4i 187 1 (V ⊆ 𝐴𝐴 = V)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  Vcvv 2737  wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739  df-in 3135  df-ss 3142
This theorem is referenced by:  vdif0im  3488
  Copyright terms: Public domain W3C validator