Theorem intnex 5257

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 5256	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	necon1bbii 2992	. . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅)
3		inteq 4879	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
4		int0 4890	. . . 4 ⊢ ∩ ∅ = V
5	3, 4	eqtrdi 2795	. . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
6	2, 5	sylbi 216	. 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V)
7		vprc 5234	. . 3 ⊢ ¬ V ∈ V
8		eleq1 2826	. . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V))
9	7, 8	mtbiri 326	. 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
10	6, 9	impbii 208	1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-int 4877

This theorem is referenced by:  intabs  5261  relintabex  41078  aiotavb  44469