Theorem int0 3876

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 3441	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	pm2.21i 647	. . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
3	2	ax-gen 1460	. . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
4		equid 1712	. . . 4 ⊢ 𝑥 = 𝑥
5	3, 4	2th 174	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥)
6	5	abbii 2305	. 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥}
7		df-int 3863	. 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)}
8		df-v 2754	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2220	1 ⊢ ∩ ∅ = V

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2160  {cab 2175  Vcvv 2752  ∅c0 3437  ∩ cint 3862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438  df-int 3863

This theorem is referenced by:  rint0  3901  intexr  4171  fiintim  6961  elfi2  7005  fi0  7008  bj-intexr  15146