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Theorem zfnuleu 3872
Description: Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2022 to strengthen the hypothesis in the form of axnul 3873). (Contributed by NM, 22-Dec-2007.)
Hypothesis
Ref Expression
zfnuleu.1 xy ¬ y x
Assertion
Ref Expression
zfnuleu ∃!xy ¬ y x
Distinct variable group:   x,y

Proof of Theorem zfnuleu
StepHypRef Expression
1 zfnuleu.1 . . . 4 xy ¬ y x
2 nbfal 1253 . . . . . 6 y x ↔ (y x ↔ ⊥ ))
32albii 1356 . . . . 5 (y ¬ y xy(y x ↔ ⊥ ))
43exbii 1493 . . . 4 (xy ¬ y xxy(y x ↔ ⊥ ))
51, 4mpbi 133 . . 3 xy(y x ↔ ⊥ )
6 nfv 1418 . . . 4 x
76bm1.1 2022 . . 3 (xy(y x ↔ ⊥ ) → ∃!xy(y x ↔ ⊥ ))
85, 7ax-mp 7 . 2 ∃!xy(y x ↔ ⊥ )
93eubii 1906 . 2 (∃!xy ¬ y x∃!xy(y x ↔ ⊥ ))
108, 9mpbir 134 1 ∃!xy ¬ y x
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wal 1240  wfal 1247  wex 1378  ∃!weu 1897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900
This theorem is referenced by: (None)
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