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Theorem int0 3732
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 3314 . . . . . 6 ¬ 𝑦 ∈ ∅
21pm2.21i 615 . . . . 5 (𝑦 ∈ ∅ → 𝑥𝑦)
32ax-gen 1393 . . . 4 𝑦(𝑦 ∈ ∅ → 𝑥𝑦)
4 equid 1645 . . . 4 𝑥 = 𝑥
53, 42th 173 . . 3 (∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦) ↔ 𝑥 = 𝑥)
65abbii 2215 . 2 {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)} = {𝑥𝑥 = 𝑥}
7 df-int 3719 . 2 ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)}
8 df-v 2643 . 2 V = {𝑥𝑥 = 𝑥}
96, 7, 83eqtr4i 2130 1 ∅ = V
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1297   = wceq 1299  wcel 1448  {cab 2086  Vcvv 2641  c0 3310   cint 3718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-nul 3311  df-int 3719
This theorem is referenced by:  rint0  3757  intexr  4015  fiintim  6746  bj-intexr  12687
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