Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xp01disjl | Structured version Visualization version GIF version |
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 8119 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 3070 | . 2 ⊢ ∅ ≠ 1o |
3 | disjsn2 4648 | . 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
4 | xpdisj1 6018 | . 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅) | |
5 | 2, 3, 4 | mp2b 10 | 1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 3016 ∩ cin 3935 ∅c0 4291 {csn 4567 × cxp 5553 1oc1o 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-xp 5561 df-rel 5562 df-suc 6197 df-1o 8102 |
This theorem is referenced by: undjudom 9593 endjudisj 9594 djuen 9595 dju1dif 9598 dju1p1e2 9599 djucomen 9603 djuassen 9604 xpdjuen 9605 mapdjuen 9606 djudom1 9608 infdju1 9615 bj-2upln1upl 34339 |
Copyright terms: Public domain | W3C validator |