Theorem xp01disjl 8353

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 8349	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2996	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4652	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4		xpdisj1 6079	. 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
5	2, 3, 4	mp2b 10	1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Syntax hints:   = wceq 1539   ≠ wne 2941   ∩ cin 3891  ∅c0 4262  {csn 4565   × cxp 5598  1_oc1o 8321

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-opab 5144  df-xp 5606  df-rel 5607  df-suc 6287  df-1o 8328

This theorem is referenced by:  undjudom  9973  endjudisj  9974  djuen  9975  dju1dif  9978  dju1p1e2  9979  djucomen  9983  djuassen  9984  xpdjuen  9985  mapdjuen  9986  djudom1  9988  infdju1  9995  bj-2upln1upl  35262