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Theorem eq0rdv 3288
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 559 . . 3 (𝜑 → (𝑥𝐴𝑥 ∈ ∅))
32ssrdv 2978 . 2 (𝜑𝐴 ⊆ ∅)
4 ss0 3284 . 2 (𝐴 ⊆ ∅ → 𝐴 = ∅)
53, 4syl 14 1 (𝜑𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1259  wcel 1409  wss 2944  c0 3251
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-in 2951  df-ss 2958  df-nul 3252
This theorem is referenced by:  nfvres  5233  snon0  6386  fzdisj  9017
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