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Theorem xp01disjl 6453
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
Assertion
Ref Expression
xp01disjl (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Proof of Theorem xp01disjl
StepHypRef Expression
1 1n0 6451 . . 3 1o ≠ ∅
21necomi 2445 . 2 ∅ ≠ 1o
3 disjsn2 3670 . 2 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4 xpdisj1 5068 . 2 (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
52, 3, 4mp2b 8 1 (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wne 2360  cin 3143  c0 3437  {csn 3607   × cxp 4639  1oc1o 6428
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-suc 4386  df-xp 4647  df-rel 4648  df-1o 6435
This theorem is referenced by:  djucomen  7234  djuassen  7235  xpdjuen  7236
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