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Theorem dfepfr 5431
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 5411 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅))
2 epel 5360 . . . . . . . 8 (𝑧 E 𝑦𝑧𝑦)
32rabbii 3418 . . . . . . 7 {𝑧𝑥𝑧 E 𝑦} = {𝑧𝑥𝑧𝑦}
4 dfin5 3869 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
53, 4eqtr4i 2821 . . . . . 6 {𝑧𝑥𝑧 E 𝑦} = (𝑥𝑦)
65eqeq1i 2799 . . . . 5 ({𝑧𝑥𝑧 E 𝑦} = ∅ ↔ (𝑥𝑦) = ∅)
76rexbii 3210 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
87imbi2i 337 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
98albii 1802 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
101, 9bitri 276 1 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1520   = wceq 1522  wne 2983  wrex 3105  {crab 3108  cin 3860  wss 3861  c0 4213   class class class wbr 4964   E cep 5355   Fr wfr 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-br 4965  df-opab 5027  df-eprel 5356  df-fr 5405
This theorem is referenced by:  onfr  6108  zfregfr  8917  onfrALTlem3  40430  onfrALT  40435  onfrALTlem3VD  40773  onfrALTVD  40777
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