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Theorem zfregfr 9599
Description: The membership relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
Assertion
Ref Expression
zfregfr E Fr 𝐴

Proof of Theorem zfregfr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfepfr 5661 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
2 vex 3478 . . . . 5 𝑥 ∈ V
3 zfreg 9589 . . . . 5 ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑦𝑥) = ∅)
42, 3mpan 688 . . . 4 (𝑥 ≠ ∅ → ∃𝑦𝑥 (𝑦𝑥) = ∅)
5 incom 4201 . . . . . 6 (𝑦𝑥) = (𝑥𝑦)
65eqeq1i 2737 . . . . 5 ((𝑦𝑥) = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3094 . . . 4 (∃𝑦𝑥 (𝑦𝑥) = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
84, 7sylib 217 . . 3 (𝑥 ≠ ∅ → ∃𝑦𝑥 (𝑥𝑦) = ∅)
98adantl 482 . 2 ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅)
101, 9mpgbir 1801 1 E Fr 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wrex 3070  Vcvv 3474  cin 3947  wss 3948  c0 4322   E cep 5579   Fr wfr 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-reg 9586
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-eprel 5580  df-fr 5631
This theorem is referenced by:  en2lp  9600  dford2  9614  noinfep  9654  zfregs  9726  bnj852  33927  dford5reg  34749  trelpss  43204
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