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Theorem epfrc 5024
 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 5004 . 2 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
3 dfin5 3548 . . . . 5 (𝐵𝑥) = {𝑦𝐵𝑦𝑥}
4 epel 4952 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
54a1i 11 . . . . . 6 (𝑦𝐵 → (𝑦 E 𝑥𝑦𝑥))
65rabbiia 3161 . . . . 5 {𝑦𝐵𝑦 E 𝑥} = {𝑦𝐵𝑦𝑥}
73, 6eqtr4i 2635 . . . 4 (𝐵𝑥) = {𝑦𝐵𝑦 E 𝑥}
87eqeq1i 2615 . . 3 ((𝐵𝑥) = ∅ ↔ {𝑦𝐵𝑦 E 𝑥} = ∅)
98rexbii 3023 . 2 (∃𝑥𝐵 (𝐵𝑥) = ∅ ↔ ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
102, 9sylibr 223 1 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   E cep 4947   Fr wfr 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997 This theorem is referenced by:  wefrc  5032  onfr  5680  epfrs  8490
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