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Theorem int0 3697
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 3288 . . . . . 6 ¬ 𝑦 ∈ ∅
21pm2.21i 610 . . . . 5 (𝑦 ∈ ∅ → 𝑥𝑦)
32ax-gen 1383 . . . 4 𝑦(𝑦 ∈ ∅ → 𝑥𝑦)
4 equid 1634 . . . 4 𝑥 = 𝑥
53, 42th 172 . . 3 (∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦) ↔ 𝑥 = 𝑥)
65abbii 2203 . 2 {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)} = {𝑥𝑥 = 𝑥}
7 df-int 3684 . 2 ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)}
8 df-v 2621 . 2 V = {𝑥𝑥 = 𝑥}
96, 7, 83eqtr4i 2118 1 ∅ = V
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  wcel 1438  {cab 2074  Vcvv 2619  c0 3284   cint 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-int 3684
This theorem is referenced by:  rint0  3722  intexr  3978  bj-intexr  11456
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