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Theorem dfepfr 5661
Description: An alternate way of saying that the membership relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dfepfr ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfepfr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffr2 5640 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅))
2 epel 5583 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
32rabbii 3438 . . . . . . 7 {𝑧𝑥𝑧 E 𝑦} = {𝑧𝑥𝑧𝑦}
4 dfin5 3956 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
53, 4eqtr4i 2763 . . . . . 6 {𝑧𝑥𝑧 E 𝑦} = (𝑥𝑦)
65eqeq1i 2737 . . . . 5 ({𝑧𝑥𝑧 E 𝑦} = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3094 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
87imbi2i 335 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
98albii 1821 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
101, 9bitri 274 1 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wne 2940  wrex 3070  {crab 3432  cin 3947  wss 3948  c0 4322   class class class wbr 5148   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-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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-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:  onfr  6403  zfregfr  9599  onfrALTlem3  43295  onfrALT  43300  onfrALTlem3VD  43638  onfrALTVD  43642
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