Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfregfr Structured version   Visualization version   GIF version

Theorem zfregfr 8622
 Description: The epsilon 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 5203 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
2 vex 3307 . . . . 5 𝑥 ∈ V
3 zfreg 8616 . . . . 5 ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑦𝑥) = ∅)
42, 3mpan 708 . . . 4 (𝑥 ≠ ∅ → ∃𝑦𝑥 (𝑦𝑥) = ∅)
5 incom 3913 . . . . . 6 (𝑦𝑥) = (𝑥𝑦)
65eqeq1i 2729 . . . . 5 ((𝑦𝑥) = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3143 . . . 4 (∃𝑦𝑥 (𝑦𝑥) = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
84, 7sylib 208 . . 3 (𝑥 ≠ ∅ → ∃𝑦𝑥 (𝑥𝑦) = ∅)
98adantl 473 . 2 ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅)
101, 9mpgbir 1839 1 E Fr 𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1596   ∈ wcel 2103   ≠ wne 2896  ∃wrex 3015  Vcvv 3304   ∩ cin 3679   ⊆ wss 3680  ∅c0 4023   E cep 5132   Fr wfr 5174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-reg 8613 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-eprel 5133  df-fr 5177 This theorem is referenced by:  en2lp  8623  dford2  8630  noinfep  8670  zfregs  8721  bnj852  31219  dford5reg  31913  trelpss  39078
 Copyright terms: Public domain W3C validator