Theorem intnex 5234

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 5233	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	necon1bbii 3065	. . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅)
3		inteq 4872	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
4		int0 4883	. . . 4 ⊢ ∩ ∅ = V
5	3, 4	syl6eq 2872	. . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
6	2, 5	sylbi 219	. 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V)
7		vprc 5212	. . 3 ⊢ ¬ V ∈ V
8		eleq1 2900	. . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V))
9	7, 8	mtbiri 329	. 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
10	6, 9	impbii 211	1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1533   ∈ wcel 2110  Vcvv 3495  ∅c0 4291  ∩ cint 4869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-int 4870

This theorem is referenced by:  intabs  5238  relintabex  39934  aiotavb  43283