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

Theorem dfepfr 5059
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 5039 . 2 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅))
2 epel 4988 . . . . . . . . 9 (𝑧 E 𝑦𝑧𝑦)
32a1i 11 . . . . . . . 8 (𝑧𝑥 → (𝑧 E 𝑦𝑧𝑦))
43rabbiia 3173 . . . . . . 7 {𝑧𝑥𝑧 E 𝑦} = {𝑧𝑥𝑧𝑦}
5 dfin5 3563 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
64, 5eqtr4i 2646 . . . . . 6 {𝑧𝑥𝑧 E 𝑦} = (𝑥𝑦)
76eqeq1i 2626 . . . . 5 ({𝑧𝑥𝑧 E 𝑦} = ∅ ↔ (𝑥𝑦) = ∅)
87rexbii 3034 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅ ↔ ∃𝑦𝑥 (𝑥𝑦) = ∅)
98imbi2i 326 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
109albii 1744 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
111, 10bitri 264 1 ( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wne 2790  wrex 2908  {crab 2911  cin 3554  wss 3555  c0 3891   class class class wbr 4613   E cep 4983   Fr wfr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-eprel 4985  df-fr 5033
This theorem is referenced by:  onfr  5722  zfregfr  8453  onfrALTlem3  38238  onfrALT  38243  onfrALTlem3VD  38603  onfrALTVD  38607
  Copyright terms: Public domain W3C validator