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Theorem zfnuleu 4714
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2595 to strengthen the hypothesis in the form of axnul 4716). (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 1486 . . . . . 6 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
32albii 1737 . . . . 5 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
43exbii 1764 . . . 4 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 ↔ ⊥))
51, 4mpbi 219 . . 3 𝑥𝑦(𝑦𝑥 ↔ ⊥)
6 nfv 1830 . . . 4 𝑥
76bm1.1 2595 . . 3 (∃𝑥𝑦(𝑦𝑥 ↔ ⊥) → ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
85, 7ax-mp 5 . 2 ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥)
93eubii 2480 . 2 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
108, 9mpbir 220 1 ∃!𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wal 1473  wfal 1480  wex 1695  ∃!weu 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463
This theorem is referenced by: (None)
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