Theorem int0 3942

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 3498	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	pm2.21i 651	. . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
3	2	ax-gen 1497	. . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
4		equid 1749	. . . 4 ⊢ 𝑥 = 𝑥
5	3, 4	2th 174	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥)
6	5	abbii 2347	. 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥}
7		df-int 3929	. 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)}
8		df-v 2804	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2262	1 ⊢ ∩ ∅ = V

Syntax hints:   → wi 4  ∀wal 1395   = wceq 1397   ∈ wcel 2202  {cab 2217  Vcvv 2802  ∅c0 3494  ∩ cint 3928

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495  df-int 3929

This theorem is referenced by:  rint0  3967  intexr  4240  fiintim  7123  elfi2  7171  fi0  7174  bj-intexr  16524