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Theorem dfepfr 5673
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 5650 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅))
2 epel 5592 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
32rabbii 3439 . . . . . . 7 {𝑧𝑥𝑧 E 𝑦} = {𝑧𝑥𝑧𝑦}
4 dfin5 3971 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
53, 4eqtr4i 2766 . . . . . 6 {𝑧𝑥𝑧 E 𝑦} = (𝑥𝑦)
65eqeq1i 2740 . . . . 5 ({𝑧𝑥𝑧 E 𝑦} = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3092 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
87imbi2i 336 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
98albii 1816 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
101, 9bitri 275 1 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wne 2938  wrex 3068  {crab 3433  cin 3962  wss 3963  c0 4339   class class class wbr 5148   E cep 5588   Fr wfr 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641
This theorem is referenced by:  onfr  6425  zfregfr  9643  onfrALTlem3  44542  onfrALT  44547  onfrALTlem3VD  44885  onfrALTVD  44889
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