Theorem intex 1986

Description: The intersection of a non-empty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse.

Assertion

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

Proof of Theorem intex

Step	Hyp	Ref	Expression
1		n0 1714	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
2		intss1 1979	. . . . 5 ⊢ (x ∈ A → ∩A ⊆ x)
3		visset 1350	. . . . . 6 ⊢ x ∈ V
4	3	ssex 1700	. . . . 5 ⊢ (∩A ⊆ x → ∩A ∈ V)
5	2, 4	syl 12	. . . 4 ⊢ (x ∈ A → ∩A ∈ V)
6	5	19.23aiv 952	. . 3 ⊢ (∃x x ∈ A → ∩A ∈ V)
7	1, 6	sylbi 174	. 2 ⊢ (¬ A = ∅ → ∩A ∈ V)
8		nvelv 1483	. . . 4 ⊢ ¬ V ∈ V
9		inteq 1968	. . . . . 6 ⊢ (A = ∅ → ∩A = ∩∅)
10		int0 1978	. . . . . 6 ⊢ ∩∅ = V
11	9, 10	syl6eq 1140	. . . . 5 ⊢ (A = ∅ → ∩A = V)
12	11	eleq1d 1155	. . . 4 ⊢ (A = ∅ → (∩A ∈ V ↔ V ∈ V))
13	8, 12	mtbiri 539	. . 3 ⊢ (A = ∅ → ¬ ∩A ∈ V)
14	13	con2i 89	. 2 ⊢ (∩A ∈ V → ¬ A = ∅)
15	7, 14	impbi 139	1 ⊢ (¬ A = ∅ ↔ ∩A ∈ V)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∩cint 1965

This theorem is referenced by:  intexab 1987  onint0 2262  onintrab 2268  onmindif2 2313  tz9.12lem1 3503  tz9.12lem3 3505  rankval 3512  oncardon 3627  oncardid 3628  cardon 3634  cardid 3635  cardcf 3706  hsupval2t 5301  hsupclt 5308

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-int 1966

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