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Theorem int0 3656
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 3255 . . . . . 6 ¬ 𝑦 ∈ ∅
21pm2.21i 585 . . . . 5 (𝑦 ∈ ∅ → 𝑥𝑦)
32ax-gen 1354 . . . 4 𝑦(𝑦 ∈ ∅ → 𝑥𝑦)
4 equid 1605 . . . 4 𝑥 = 𝑥
53, 42th 167 . . 3 (∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦) ↔ 𝑥 = 𝑥)
65abbii 2169 . 2 {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)} = {𝑥𝑥 = 𝑥}
7 df-int 3643 . 2 ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)}
8 df-v 2576 . 2 V = {𝑥𝑥 = 𝑥}
96, 7, 83eqtr4i 2086 1 ∅ = V
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257   = wceq 1259  wcel 1409  {cab 2042  Vcvv 2574  c0 3251   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-nul 3252  df-int 3643
This theorem is referenced by:  rint0  3681  intexr  3931  bj-intexr  10387
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