Theorem xp01disjl 8548

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

Step	Hyp	Ref	Expression
1		1n0 8544	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3001	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4737	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4		xpdisj1 6192	. 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
5	2, 3, 4	mp2b 10	1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Syntax hints:   = wceq 1537   ≠ wne 2946   ∩ cin 3975  ∅c0 4352  {csn 4648   × cxp 5698  1_oc1o 8515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707  df-suc 6401  df-1o 8522

This theorem is referenced by:  undjudom  10237  endjudisj  10238  djuen  10239  dju1dif  10242  dju1p1e2  10243  djucomen  10247  djuassen  10248  xpdjuen  10249  mapdjuen  10250  djudom1  10252  infdju1  10259  bj-2upln1upl  36990