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Theorem epfrc 5537
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 5517 . 2 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
3 dfin5 3874 . . . . 5 (𝐵𝑥) = {𝑦𝐵𝑦𝑥}
4 epel 5463 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
54rabbii 3383 . . . . 5 {𝑦𝐵𝑦 E 𝑥} = {𝑦𝐵𝑦𝑥}
63, 5eqtr4i 2768 . . . 4 (𝐵𝑥) = {𝑦𝐵𝑦 E 𝑥}
76eqeq1i 2742 . . 3 ((𝐵𝑥) = ∅ ↔ {𝑦𝐵𝑦 E 𝑥} = ∅)
87rexbii 3170 . 2 (∃𝑥𝐵 (𝐵𝑥) = ∅ ↔ ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
92, 8sylibr 237 1 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wrex 3062  {crab 3065  Vcvv 3408  cin 3865  wss 3866  c0 4237   class class class wbr 5053   E cep 5459   Fr wfr 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-eprel 5460  df-fr 5509
This theorem is referenced by:  wefrc  5545  onfr  6252  epfrs  9347
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