Theorem intexr 4234

Description: If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
intexr	⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)

Proof of Theorem intexr

Step	Hyp	Ref	Expression
1		vprc 4216	. . 3 ⊢ ¬ V ∈ V
2		inteq 3926	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
3		int0 3937	. . . . 5 ⊢ ∩ ∅ = V
4	2, 3	eqtrdi 2278	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
5	4	eleq1d 2298	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V))
6	1, 5	mtbiri 679	. 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V)
7	6	necon2ai 2454	1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  Vcvv 2799  ∅c0 3491  ∩ cint 3923

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-nul 3492  df-int 3924

This theorem is referenced by:  fival  7145