Theorem intnex 5011

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 5010	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	necon1bbii 3018	. . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅)
3		inteq 4668	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
4		int0 4679	. . . 4 ⊢ ∩ ∅ = V
5	3, 4	syl6eq 2847	. . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
6	2, 5	sylbi 209	. 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V)
7		vprc 4990	. . 3 ⊢ ¬ V ∈ V
8		eleq1 2864	. . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V))
9	7, 8	mtbiri 319	. 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
10	6, 9	impbii 201	1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1653   ∈ wcel 2157  Vcvv 3383  ∅c0 4113  ∩ cint 4665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-v 3385  df-dif 3770  df-in 3774  df-ss 3781  df-nul 4114  df-int 4666

This theorem is referenced by:  intabs  5015  relintabex  38658  aiotavb  41927