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Theorem nel0 3468
Description: From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
Hypothesis
Ref Expression
nel0.1 ¬ 𝑥𝐴
Assertion
Ref Expression
nel0 𝐴 = ∅
Distinct variable group:   𝑥,𝐴

Proof of Theorem nel0
StepHypRef Expression
1 eq0 3465 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 nel0.1 . 2 ¬ 𝑥𝐴
31, 2mpgbir 1464 1 𝐴 = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1364  wcel 2164  c0 3446
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-nul 3447
This theorem is referenced by: (None)
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