Theorem int0 3845

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

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		noel 3418	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	pm2.21i 641	. . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
3	2	ax-gen 1442	. . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
4		equid 1694	. . . 4 ⊢ 𝑥 = 𝑥
5	3, 4	2th 173	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥)
6	5	abbii 2286	. 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥}
7		df-int 3832	. 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)}
8		df-v 2732	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2201	1 ⊢ ∩ ∅ = V

Syntax hints:   → wi 4  ∀wal 1346   = wceq 1348   ∈ wcel 2141  {cab 2156  Vcvv 2730  ∅c0 3414  ∩ cint 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-int 3832

This theorem is referenced by:  rint0  3870  intexr  4136  fiintim  6906  elfi2  6949  fi0  6952  bj-intexr  13943