Theorem int0 4425

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4028	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4417	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 220	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 253	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2607	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874  ∩ cint 4410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875  df-int 4411

This theorem is referenced by:  unissint  4436  uniintsn  4449  rint0  4452  intex  4747  intnex  4748  oev2  7490  fiint  8122  elfi2  8203  fi0  8209  cardmin2  8707  00lsp  18802  cmpfi  21021  ptbasfi  21194  fbssint  21452  fclscmp  21644  rankeq1o  31448  heibor1lem  32778