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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-xpima1sn | Structured version Visualization version GIF version |
Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6157 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-xpima1sn | ⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-xpimasn 35533 | . 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) | |
2 | iffalse 4515 | . 2 ⊢ (¬ 𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = ∅) | |
3 | 1, 2 | eqtrid 2783 | 1 ⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2106 ∅c0 4302 ifcif 4506 {csn 4606 × cxp 5651 “ cima 5656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pr 5404 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-br 5126 df-opab 5188 df-xp 5659 df-rel 5660 df-cnv 5661 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 |
This theorem is referenced by: bj-projval 35574 |
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