Mathbox for Glauco Siliprandi |
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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 5083 | . . . 4 ⊢ Disj 𝑗 ∈ 𝐴 {𝑗} | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 {𝑗}) |
3 | 2 | disjxp1 42945 | . 2 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
4 | 3 | mptru 1547 | 1 ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1541 {csn 4573 Disj wdisj 5057 × cxp 5618 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rmo 3349 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-disj 5058 df-opab 5155 df-xp 5626 df-rel 5627 |
This theorem is referenced by: sge0xp 44312 |
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