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| Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.) | 
| Ref | Expression | 
|---|---|
| xp01disjl | ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1n0 8526 | . . 3 ⊢ 1o ≠ ∅ | |
| 2 | 1 | necomi 2995 | . 2 ⊢ ∅ ≠ 1o | 
| 3 | disjsn2 4712 | . 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
| 4 | xpdisj1 6181 | . 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅) | |
| 5 | 2, 3, 4 | mp2b 10 | 1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 {csn 4626 × cxp 5683 1oc1o 8499 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 df-suc 6390 df-1o 8506 | 
| This theorem is referenced by: undjudom 10208 endjudisj 10209 djuen 10210 dju1dif 10213 dju1p1e2 10214 djucomen 10218 djuassen 10219 xpdjuen 10220 mapdjuen 10221 djudom1 10223 infdju1 10230 bj-2upln1upl 37025 | 
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