Theorem vn0 3231

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)

Assertion

Ref	Expression
vn0	⊢ V ≠ ∅

Proof of Theorem vn0

Step	Hyp	Ref	Expression
1		vex 2560	. 2 ⊢ 𝑥 ∈ V
2		ne0i 3230	. 2 ⊢ (𝑥 ∈ V → V ≠ ∅)
3	1, 2	ax-mp 7	1 ⊢ V ≠ ∅

Syntax hints:   ∈ wcel 1393   ≠ wne 2204  Vcvv 2557  ∅c0 3224

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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-nul 3225

This theorem is referenced by: (None)