Theorem xp01disjl 8504

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 8500	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2986	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4688	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4		xpdisj1 6150	. 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
5	2, 3, 4	mp2b 10	1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Syntax hints:   = wceq 1540   ≠ wne 2932   ∩ cin 3925  ∅c0 4308  {csn 4601   × cxp 5652  1_oc1o 8473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-opab 5182  df-xp 5660  df-rel 5661  df-suc 6358  df-1o 8480

This theorem is referenced by:  undjudom  10182  endjudisj  10183  djuen  10184  dju1dif  10187  dju1p1e2  10188  djucomen  10192  djuassen  10193  xpdjuen  10194  mapdjuen  10195  djudom1  10197  infdju1  10204  bj-2upln1upl  37042