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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.
StepHypRef Expression
1 noel 3441 . . . . . 6 ¬ 𝑦 ∈ ∅
21pm2.21i 647 . . . . 5 (𝑦 ∈ ∅ → 𝑥𝑦)
32ax-gen 1460 . . . 4 𝑦(𝑦 ∈ ∅ → 𝑥𝑦)
4 equid 1712 . . . 4 𝑥 = 𝑥
53, 42th 174 . . 3 (∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦) ↔ 𝑥 = 𝑥)
65abbii 2305 . 2 {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)} = {𝑥𝑥 = 𝑥}
7 df-int 3863 . 2 ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)}
8 df-v 2754 . 2 V = {𝑥𝑥 = 𝑥}
96, 7, 83eqtr4i 2220 1 ∅ = V
Colors of variables: wff set class
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
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