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Theorem eldifsni 3815
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 3813 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵𝐴𝐶))
21simprbi 275 1 (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2203  wne 2412  cdif 3207  {csn 3682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214
This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2814  df-dif 3212  df-sn 3688
This theorem is referenced by:  neldifsn  3816  suppssov1  6254  suppssfvg  6454  elfi2  7250  fiuni  7256  fifo  7258  en2other2  7490  oddprm  12935  ringelnzr  14306  lgslem1  15843  lgseisenlem2  15914  lgseisenlem4  15916  lgseisen  15917  lgsquadlem1  15920  lgsquad2  15926  m1lgs  15928
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