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 5049 | . . . 4 ⊢ Disj 𝑗 ∈ 𝐴 {𝑗} | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 {𝑗}) |
3 | 2 | disjxp1 41324 | . 2 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
4 | 3 | mptru 1540 | 1 ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1534 {csn 4560 Disj wdisj 5023 × cxp 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-disj 5024 df-opab 5121 df-xp 5555 df-rel 5556 |
This theorem is referenced by: sge0xp 42705 |
Copyright terms: Public domain | W3C validator |