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Theorem zfnuleu 4612
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2499 to strengthen the hypothesis in the form of axnul 4614). (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 1485 . . . . . 6 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
32albii 1722 . . . . 5 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
43exbii 1752 . . . 4 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 ↔ ⊥))
51, 4mpbi 218 . . 3 𝑥𝑦(𝑦𝑥 ↔ ⊥)
6 nfv 1796 . . . 4 𝑥
76bm1.1 2499 . . 3 (∃𝑥𝑦(𝑦𝑥 ↔ ⊥) → ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
85, 7ax-mp 5 . 2 ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥)
93eubii 2384 . 2 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
108, 9mpbir 219 1 ∃!𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wal 1472  wfal 1479  wex 1694  ∃!weu 2362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367
This theorem is referenced by: (None)
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