Theorem intnex 2735

Description: If a class intersection is not a set, it must be the universe.

Assertion

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

Proof of Theorem intnex

Step	Hyp	Ref	Expression
1		intex 2734	. . . 4 ⊢ (A ≠ ∅ ↔ ∩A ∈ V)
2	1	necon1bbii 1620	. . 3 ⊢ (¬ ∩A ∈ V ↔ A = ∅)
3		inteq 2540	. . . 4 ⊢ (A = ∅ → ∩A = ∩∅)
4		int0 2551	. . . 4 ⊢ ∩∅ = V
5	3, 4	syl6eq 1526	. . 3 ⊢ (A = ∅ → ∩A = V)
6	2, 5	sylbi 199	. 2 ⊢ (¬ ∩A ∈ V → ∩A = V)
7		nvelv 2718	. . 3 ⊢ ¬ V ∈ V
8		eleq1 1537	. . 3 ⊢ (∩A = V → (∩A ∈ V ↔ V ∈ V))
9	7, 8	mtbiri 719	. 2 ⊢ (∩A = V → ¬ ∩A ∈ V)
10	6, 9	impbi 157	1 ⊢ (¬ ∩A ∈ V ↔ ∩A = V)

Syntax hints:  ¬ wn 2   ↔ wb 146   = wceq 958   ∈ wcel 960  Vcvv 1814  ∅c0 2283  ∩cint 2537

This theorem is referenced by:  intabs 2738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-int 2538

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