Theorem xp01disjl 8461

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 8456	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3011	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4671	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4		xpdisj1 6146	. 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
5	2, 3, 4	mp2b 10	1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Syntax hints:   = wceq 1560   ≠ wne 2957   ∩ cin 3903  ∅c0 4285  {csn 4582   × cxp 5645  1_oc1o 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-rel 5654  df-suc 6352  df-1o 8437

This theorem is referenced by:  undjudom  10124  endjudisj  10125  djuen  10126  dju1dif  10129  dju1p1e2  10130  djucomen  10134  djuassen  10135  xpdjuen  10136  mapdjuen  10137  djudom1  10139  infdju1  10146  bj-2upln1upl  37509