Theorem 0iin 4023

Description: An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
0iin	⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

Proof of Theorem 0iin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3967	. 2 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
2		vex 2802	. . . 4 ⊢ 𝑦 ∈ V
3		ral0 3593	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴
4	2, 3	2th 174	. . 3 ⊢ (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
5	4	abbi2i 2344	. 2 ⊢ V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝐴}
6	1, 5	eqtr4i 2253	1 ⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V

Syntax hints:   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  Vcvv 2799  ∅c0 3491  ∩ ciin 3965

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-nul 3492  df-iin 3967

This theorem is referenced by:  riin0  4036  iin0r  4252