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Theorem zfnuleu 4920
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2756 to strengthen the hypothesis in the form of axnul 4922). (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 1643 . . . . . 6 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
32albii 1895 . . . . 5 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
43exbii 1924 . . . 4 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 ↔ ⊥))
51, 4mpbi 220 . . 3 𝑥𝑦(𝑦𝑥 ↔ ⊥)
6 nfv 1995 . . . 4 𝑥
76bm1.1 2756 . . 3 (∃𝑥𝑦(𝑦𝑥 ↔ ⊥) → ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
85, 7ax-mp 5 . 2 ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥)
93eubii 2640 . 2 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
108, 9mpbir 221 1 ∃!𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1629  wfal 1636  wex 1852  ∃!weu 2618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623
This theorem is referenced by: (None)
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