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Theorem vss 3241
 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 2942 . . 3 A ⊆ V
21biantrur 287 . 2 (V ⊆ A ↔ (A ⊆ V V ⊆ A))
3 eqss 2937 . 2 (A = V ↔ (A ⊆ V V ⊆ A))
42, 3bitr4i 176 1 (V ⊆ AA = V)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228  Vcvv 2535   ⊆ wss 2894 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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537  df-in 2901  df-ss 2908 This theorem is referenced by:  vdif0im  3264
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