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Theorem dif1o 5901
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 5893 . . . 4 1𝑜 = {∅}
21difeq2i 3037 . . 3 (B ∖ 1𝑜) = (B ∖ {∅})
32eleq2i 2086 . 2 (A (B ∖ 1𝑜) ↔ A (B ∖ {∅}))
4 eldifsn 3446 . 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 1375  wne 2186  cdif 2892  c0 3202  {csn 3327  1𝑜c1o 5874
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  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 2898  df-un 2900  df-nul 3203  df-sn 3333  df-suc 4031  df-1o 5881
This theorem is referenced by: (None)
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