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Theorem vss 3258
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 2959 . . 3 A ⊆ V
21biantrur 287 . 2 (V ⊆ A ↔ (A ⊆ V V ⊆ A))
3 eqss 2954 . 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 1242  Vcvv 2551  wss 2911
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by:  vdif0im  3281
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