Theorem int0 3940

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 3554	. . . . . 6 ⊢ ¬ y ∈ ∅
2	1	pm2.21i 123	. . . . 5 ⊢ (y ∈ ∅ → x ∈ y)
3	2	ax-gen 1546	. . . 4 ⊢ ∀y(y ∈ ∅ → x ∈ y)
4		eqid 2353	. . . 4 ⊢ x = x
5	3, 4	2th 230	. . 3 ⊢ (∀y(y ∈ ∅ → x ∈ y) ↔ x = x)
6	5	abbii 2465	. 2 ⊢ {x ∣ ∀y(y ∈ ∅ → x ∈ y)} = {x ∣ x = x}
7		df-int 3927	. 2 ⊢ ∩∅ = {x ∣ ∀y(y ∈ ∅ → x ∈ y)}
8		df-v 2861	. 2 ⊢ V = {x ∣ x = x}
9	6, 7, 8	3eqtr4i 2383	1 ⊢ ∩∅ = V

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∅c0 3550  ∩cint 3926

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-int 3927

This theorem is referenced by:  unissint  3950  uniintsn  3963  rint0  3966