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Theorem int0 3785
 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 3367 . . . . . 6 ¬ 𝑦 ∈ ∅
21pm2.21i 635 . . . . 5 (𝑦 ∈ ∅ → 𝑥𝑦)
32ax-gen 1425 . . . 4 𝑦(𝑦 ∈ ∅ → 𝑥𝑦)
4 equid 1677 . . . 4 𝑥 = 𝑥
53, 42th 173 . . 3 (∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦) ↔ 𝑥 = 𝑥)
65abbii 2255 . 2 {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)} = {𝑥𝑥 = 𝑥}
7 df-int 3772 . 2 ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)}
8 df-v 2688 . 2 V = {𝑥𝑥 = 𝑥}
96, 7, 83eqtr4i 2170 1 ∅ = V
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329   = wceq 1331   ∈ wcel 1480  {cab 2125  Vcvv 2686  ∅c0 3363  ∩ cint 3771 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364  df-int 3772 This theorem is referenced by:  rint0  3810  intexr  4075  fiintim  6817  elfi2  6860  fi0  6863  bj-intexr  13165
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