Theorem vn0 3505

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 2805	. 2 ⊢ 𝑥 ∈ V
2		ne0i 3501	. 2 ⊢ (𝑥 ∈ V → V ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ V ≠ ∅

Syntax hints:   ∈ wcel 2202   ≠ wne 2402  Vcvv 2802  ∅c0 3494

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by: (None)