Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > disjsnxp | Structured version Visualization version GIF version |
Description: The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
disjsnxp | ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sndisj 4574 | . . . 4 ⊢ Disj 𝑗 ∈ 𝐴 {𝑗} | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 {𝑗}) |
3 | 2 | disjxp1 38263 | . 2 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
4 | 3 | trud 1484 | 1 ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1476 {csn 4125 Disj wdisj 4553 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-disj 4554 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: sge0xp 39322 |
Copyright terms: Public domain | W3C validator |