Theorem xp01disjl 8401

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 8397	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2979	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4662	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4		xpdisj1 6104	. 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
5	2, 3, 4	mp2b 10	1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Syntax hints:   = wceq 1540   ≠ wne 2925   ∩ cin 3898  ∅c0 4280  {csn 4573   × cxp 5611  1_oc1o 8372

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5151  df-xp 5619  df-rel 5620  df-suc 6307  df-1o 8379

This theorem is referenced by:  undjudom  10050  endjudisj  10051  djuen  10052  dju1dif  10055  dju1p1e2  10056  djucomen  10060  djuassen  10061  xpdjuen  10062  mapdjuen  10063  djudom1  10065  infdju1  10072  bj-2upln1upl  37015