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

Theorem epfrc 5244
Description: A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)
Hypothesis
Ref Expression
epfrc.1 𝐵 ∈ V
Assertion
Ref Expression
epfrc (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem epfrc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 epfrc.1 . . 3 𝐵 ∈ V
21frc 5224 . 2 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
3 dfin5 3715 . . . . 5 (𝐵𝑥) = {𝑦𝐵𝑦𝑥}
4 epel 5174 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
54rabbii 3317 . . . . 5 {𝑦𝐵𝑦 E 𝑥} = {𝑦𝐵𝑦𝑥}
63, 5eqtr4i 2777 . . . 4 (𝐵𝑥) = {𝑦𝐵𝑦 E 𝑥}
76eqeq1i 2757 . . 3 ((𝐵𝑥) = ∅ ↔ {𝑦𝐵𝑦 E 𝑥} = ∅)
87rexbii 3171 . 2 (∃𝑥𝐵 (𝐵𝑥) = ∅ ↔ ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
92, 8sylibr 224 1 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wrex 3043  {crab 3046  Vcvv 3332  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:  wefrc  5252  onfr  5916  epfrs  8772
  Copyright terms: Public domain W3C validator