Theorem intnex 5303

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 5302	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	necon1bbii 2975	. . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅)
3		inteq 4916	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
4		int0 4929	. . . 4 ⊢ ∩ ∅ = V
5	3, 4	eqtrdi 2781	. . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
6	2, 5	sylbi 217	. 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V)
7		vprc 5273	. . 3 ⊢ ¬ V ∈ V
8		eleq1 2817	. . 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 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300  df-int 4914

This theorem is referenced by:  intabs  5307  relintabex  43577  aiotavb  47095