Theorem intexr 4284

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 4263	. . 3 ⊢ ¬ V ∈ V
2		inteq 3971	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
3		int0 3982	. . . . 5 ⊢ ∩ ∅ = V
4	2, 3	eqtrdi 2287	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
5	4	eleq1d 2307	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V))
6	1, 5	mtbiri 686	. 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V)
7	6	necon2ai 2474	1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  Vcvv 2821  ∅c0 3520  ∩ cint 3968

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4247

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-nul 3521  df-int 3969

This theorem is referenced by:  fival  7298