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Theorem 0iin 3878
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.
StepHypRef Expression
1 df-iin 3823 . 2 𝑥 ∈ ∅ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
2 vex 2692 . . . 4 𝑦 ∈ V
3 ral0 3468 . . . 4 𝑥 ∈ ∅ 𝑦𝐴
42, 32th 173 . . 3 (𝑦 ∈ V ↔ ∀𝑥 ∈ ∅ 𝑦𝐴)
54abbi2i 2255 . 2 V = {𝑦 ∣ ∀𝑥 ∈ ∅ 𝑦𝐴}
61, 5eqtr4i 2164 1 𝑥 ∈ ∅ 𝐴 = V
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  {cab 2126  wral 2417  Vcvv 2689  c0 3367   ciin 3821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-dif 3077  df-nul 3368  df-iin 3823
This theorem is referenced by:  riin0  3891  iin0r  4100
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