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Theorem dif1o 5934
Description: Two ways to say that A is a nonzero number of the set B. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
dif1o (A (B ∖ 1𝑜) ↔ (A B A ≠ ∅))

Proof of Theorem dif1o
StepHypRef Expression
1 df1o2 5926 . . . 4 1𝑜 = {∅}
21difeq2i 3034 . . 3 (B ∖ 1𝑜) = (B ∖ {∅})
32eleq2i 2086 . 2 (A (B ∖ 1𝑜) ↔ A (B ∖ {∅}))
4 eldifsn 3467 . 2 (A (B ∖ {∅}) ↔ (A B A ≠ ∅))
53, 4bitri 173 1 (A (B ∖ 1𝑜) ↔ (A B A ≠ ∅))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1374  wne 2186  cdif 2889  c0 3199  {csn 3348  1𝑜c1o 5907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rab 2291  df-v 2535  df-dif 2895  df-un 2897  df-nul 3200  df-sn 3354  df-suc 4055  df-1o 5914
This theorem is referenced by: (None)
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