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Theorem dfepfr 5243
Description: An alternate way of saying that the epsilon 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 5223 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅))
2 epel 5174 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
32rabbii 3317 . . . . . . 7 {𝑧𝑥𝑧 E 𝑦} = {𝑧𝑥𝑧𝑦}
4 dfin5 3715 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
53, 4eqtr4i 2777 . . . . . 6 {𝑧𝑥𝑧 E 𝑦} = (𝑥𝑦)
65eqeq1i 2757 . . . . 5 ({𝑧𝑥𝑧 E 𝑦} = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3171 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
87imbi2i 325 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
98albii 1888 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
101, 9bitri 264 1 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1622   = wceq 1624  wne 2924  wrex 3043  {crab 3046  cin 3706  wss 3707  c0 4050   class class class wbr 4796   E cep 5170   Fr wfr 5214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-eprel 5171  df-fr 5217
This theorem is referenced by:  onfr  5916  zfregfr  8666  onfrALTlem3  39253  onfrALT  39258  onfrALTlem3VD  39614  onfrALTVD  39618
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