Theorem xp01disjl 8513

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 8509	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2984	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4718	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4		xpdisj1 6167	. 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅)
5	2, 3, 4	mp2b 10	1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅

Syntax hints:   = wceq 1533   ≠ wne 2929   ∩ cin 3943  ∅c0 4322  {csn 4630   × cxp 5676  1_oc1o 8480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5212  df-xp 5684  df-rel 5685  df-suc 6377  df-1o 8487

This theorem is referenced by:  undjudom  10192  endjudisj  10193  djuen  10194  dju1dif  10197  dju1p1e2  10198  djucomen  10202  djuassen  10203  xpdjuen  10204  mapdjuen  10205  djudom1  10207  infdju1  10214  bj-2upln1upl  36634