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Mirrors > Home > ILE Home > Th. List > disjsnxp | 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 4014 | . . . 4 ⊢ Disj 𝑗 ∈ 𝐴 {𝑗} | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 {𝑗}) |
3 | 2 | disjxp1 6261 | . 2 ⊢ (⊤ → Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
4 | 3 | mptru 1373 | 1 ⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1365 {csn 3607 Disj wdisj 3995 × cxp 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rmo 2476 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-disj 3996 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fo 5241 df-fv 5243 df-1st 6165 |
This theorem is referenced by: fsum2dlemstep 11474 fisumcom2 11478 fprod2dlemstep 11662 fprodcom2fi 11666 |
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