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Theorem eq0rdv 3312
Description: Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
Hypothesis
Ref Expression
eq0rdv.1 (𝜑 → ¬ 𝑥𝐴)
Assertion
Ref Expression
eq0rdv (𝜑𝐴 = ∅)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem eq0rdv
StepHypRef Expression
1 eq0rdv.1 . . . 4 (𝜑 → ¬ 𝑥𝐴)
21pm2.21d 582 . . 3 (𝜑 → (𝑥𝐴𝑥 ∈ ∅))
32ssrdv 3018 . 2 (𝜑𝐴 ⊆ ∅)
4 ss0 3308 . 2 (𝐴 ⊆ ∅ → 𝐴 = ∅)
53, 4syl 14 1 (𝜑𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1287  wcel 1436  wss 2986  c0 3272
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-nul 3273
This theorem is referenced by:  exmid01  3999  dcextest  4362  nfvres  5285  map0b  6377  snon0  6573  snexxph  6586  fodjuomnilem0  6723  fzdisj  9375
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