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Theorem zfregfr 9549
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 5622 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
2 vex 3451 . . . . 5 𝑥 ∈ V
3 zfreg 9539 . . . . 5 ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑦𝑥) = ∅)
42, 3mpan 689 . . . 4 (𝑥 ≠ ∅ → ∃𝑦𝑥 (𝑦𝑥) = ∅)
5 incom 4165 . . . . . 6 (𝑦𝑥) = (𝑥𝑦)
65eqeq1i 2738 . . . . 5 ((𝑦𝑥) = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3094 . . . 4 (∃𝑦𝑥 (𝑦𝑥) = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
84, 7sylib 217 . . 3 (𝑥 ≠ ∅ → ∃𝑦𝑥 (𝑥𝑦) = ∅)
98adantl 483 . 2 ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅)
101, 9mpgbir 1802 1 E Fr 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wrex 3070  Vcvv 3447  cin 3913  wss 3914  c0 4286   E cep 5540   Fr wfr 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-eprel 5541  df-fr 5592
This theorem is referenced by:  en2lp  9550  dford2  9564  noinfep  9604  zfregs  9676  bnj852  33597  dford5reg  34420  trelpss  42827
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