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Theorem epfrc 5436
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 5416 . 2 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
3 dfin5 3873 . . . . 5 (𝐵𝑥) = {𝑦𝐵𝑦𝑥}
4 epel 5364 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
54rabbii 3421 . . . . 5 {𝑦𝐵𝑦 E 𝑥} = {𝑦𝐵𝑦𝑥}
63, 5eqtr4i 2824 . . . 4 (𝐵𝑥) = {𝑦𝐵𝑦 E 𝑥}
76eqeq1i 2802 . . 3 ((𝐵𝑥) = ∅ ↔ {𝑦𝐵𝑦 E 𝑥} = ∅)
87rexbii 3213 . 2 (∃𝑥𝐵 (𝐵𝑥) = ∅ ↔ ∃𝑥𝐵 {𝑦𝐵𝑦 E 𝑥} = ∅)
92, 8sylibr 235 1 (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wrex 3108  {crab 3111  Vcvv 3440  cin 3864  wss 3865  c0 4217   class class class wbr 4968   E cep 5359   Fr wfr 5406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-eprel 5360  df-fr 5409
This theorem is referenced by:  wefrc  5444  onfr  6112  epfrs  9026
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