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Theorem epfrc 5670
Description: A subset of a well-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 5648 . 2 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
3 dfin5 3959 . . . . 5 (𝐵𝑥) = {𝑦𝐵𝑦𝑥}
4 epel 5587 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
54rabbii 3442 . . . . 5 {𝑦𝐵𝑦 E 𝑥} = {𝑦𝐵𝑦𝑥}
63, 5eqtr4i 2768 . . . 4 (𝐵𝑥) = {𝑦𝐵𝑦 E 𝑥}
76eqeq1i 2742 . . 3 ((𝐵𝑥) = ∅ ↔ {𝑦𝐵𝑦 E 𝑥} = ∅)
87rexbii 3094 . 2 (∃𝑥𝐵 (𝐵𝑥) = ∅ ↔ ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
92, 8sylibr 234 1 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  wss 3951  c0 4333   class class class wbr 5143   E cep 5583   Fr wfr 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637
This theorem is referenced by:  wefrc  5679  onfr  6423  epfrs  9771
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