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Theorem zfnuleu 4020
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2100 to strengthen the hypothesis in the form of axnul 4021). (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 1325 . . . . . 6 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
32albii 1429 . . . . 5 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
43exbii 1567 . . . 4 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 ↔ ⊥))
51, 4mpbi 144 . . 3 𝑥𝑦(𝑦𝑥 ↔ ⊥)
6 nfv 1491 . . . 4 𝑥
76bm1.1 2100 . . 3 (∃𝑥𝑦(𝑦𝑥 ↔ ⊥) → ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
85, 7ax-mp 5 . 2 ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥)
93eubii 1984 . 2 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃!𝑥𝑦(𝑦𝑥 ↔ ⊥))
108, 9mpbir 145 1 ∃!𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1312  wfal 1319  wex 1451  ∃!weu 1975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978
This theorem is referenced by: (None)
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