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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 6207 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 36938 | . 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) | |
2 | iffalse 4540 | . 2 ⊢ (¬ 𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = ∅) | |
3 | 1, 2 | eqtrid 2787 | 1 ⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ∅c0 4339 ifcif 4531 {csn 4631 × cxp 5687 “ cima 5692 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: bj-projval 36979 |
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