Theorem int0 1978

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.

Assertion

Ref	Expression
int0	⊢ ∩∅ = V

Proof of Theorem int0

Step	Hyp	Ref	Expression
1		noel 1711	. . . . . 6 ⊢ ¬ y ∈ ∅
2	1	pm2.21i 73	. . . . 5 ⊢ (y ∈ ∅ → x ∈ y)
3	2	ax-gen 677	. . . 4 ⊢ ∀y(y ∈ ∅ → x ∈ y)
4		cleqid 1102	. . . 4 ⊢ x = x
5	3, 4	2th 540	. . 3 ⊢ (∀y(y ∈ ∅ → x ∈ y) ↔ x = x)
6	5	biabi 1181	. 2 ⊢ {x∣∀y(y ∈ ∅ → x ∈ y)} = {x∣x = x}
7		df-int 1966	. 2 ⊢ ∩∅ = {x∣∀y(y ∈ ∅ → x ∈ y)}
8		df-v 1349	. 2 ⊢ V = {x∣x = x}
9	6, 7, 8	3eqtr4 1126	1 ⊢ ∩∅ = V

Syntax hints:   → wi 2  ∀wal 672   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  ∩cint 1965

This theorem is referenced by:  intex 1986  fiint 3445

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-16 922  ax-17 925  ax-ext 1074

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-v 1349  df-dif 1489  df-nul 1708  df-int 1966

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