Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dif1o GIF version

Theorem dif1o 5960
 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 5952 . . . 4 1𝑜 = {∅}
21difeq2i 3053 . . 3 (B ∖ 1𝑜) = (B ∖ {∅})
32eleq2i 2101 . 2 (A (B ∖ 1𝑜) ↔ A (B ∖ {∅}))
4 eldifsn 3486 . 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 1390   ≠ wne 2201   ∖ cdif 2908  ∅c0 3218  {csn 3367  1𝑜c1o 5933 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-suc 4074  df-1o 5940 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator