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Theorem eldifsni 3620
Description: Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
Assertion
Ref Expression
eldifsni (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)

Proof of Theorem eldifsni
StepHypRef Expression
1 eldifsn 3618 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵𝐴𝐶))
21simprbi 271 1 (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  wne 2283  cdif 3036  {csn 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-sn 3501
This theorem is referenced by:  neldifsn  3621  suppssfv  5944  suppssov1  5945  elfi2  6826  fiuni  6832  fifo  6834  en2other2  7016
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