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Theorem xp01disj 5956
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
Assertion
Ref Expression
xp01disj ((A × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Proof of Theorem xp01disj
StepHypRef Expression
1 1n0 5955 . . 3 1𝑜 ≠ ∅
21necomi 2284 . 2 ∅ ≠ 1𝑜
3 xpsndisj 4692 . 2 (∅ ≠ 1𝑜 → ((A × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
42, 3ax-mp 7 1 ((A × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wne 2201  cin 2910  c0 3218  {csn 3367   × cxp 4286  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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-1o 5940
This theorem is referenced by:  endisj  6234
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