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Theorem zfnuleu 4777
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2605 to strengthen the hypothesis in the form of axnul 4779). (Contributed by NM, 22-Dec-2007.)
Hypothesis
Ref Expression
zfnuleu.1 𝑥𝑦 ¬ 𝑦𝑥
Assertion
Ref Expression
zfnuleu ∃!𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem zfnuleu
StepHypRef Expression
1 zfnuleu.1 . . . 4 𝑥𝑦 ¬ 𝑦𝑥
2 nbfal 1493 . . . . . 6 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
32albii 1745 . . . . 5 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
43exbii 1772 . . . 4 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 ↔ ⊥))
51, 4mpbi 220 . . 3 𝑥𝑦(𝑦𝑥 ↔ ⊥)
6 nfv 1841 . . . 4 𝑥
76bm1.1 2605 . . 3 (∃𝑥𝑦(𝑦𝑥 ↔ ⊥) → ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
85, 7ax-mp 5 . 2 ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥)
93eubii 2490 . 2 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
108, 9mpbir 221 1 ∃!𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1479  wfal 1486  wex 1702  ∃!weu 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473
This theorem is referenced by: (None)
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