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Theorem eq0rdv 3467
Description: Deduction 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 619 . . 3 (𝜑 → (𝑥𝐴𝑥 ∈ ∅))
32ssrdv 3161 . 2 (𝜑𝐴 ⊆ ∅)
4 ss0 3463 . 2 (𝐴 ⊆ ∅ → 𝐴 = ∅)
53, 4syl 14 1 (𝜑𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1353  wcel 2148  wss 3129  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  exmid01  4195  dcextest  4577  nfvres  5544  map0b  6681  snon0  6929  snexxph  6943  fodju0  7139  fzdisj  10035  bldisj  13565
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