Theorem intnex 5363

Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)

Assertion

Ref	Expression
intnex	⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V)

Proof of Theorem intnex

Step	Hyp	Ref	Expression
1		intex 5362	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	necon1bbii 2996	. . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅)
3		inteq 4973	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
4		int0 4986	. . . 4 ⊢ ∩ ∅ = V
5	3, 4	eqtrdi 2796	. . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
6	2, 5	sylbi 217	. 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V)
7		vprc 5333	. . 3 ⊢ ¬ V ∈ V
8		eleq1 2832	. . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V))
9	7, 8	mtbiri 327	. 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
10	6, 9	impbii 209	1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-nul 4353  df-int 4971

This theorem is referenced by:  intabs  5367  relintabex  43543  aiotavb  47005