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| Mirrors > Home > MPE Home > Th. List > elimasng | Structured version Visualization version GIF version | ||
| Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6105, remove, and relabel elimasng1 6105 to "elimasng". |
| Ref | Expression |
|---|---|
| elimasng | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elimasng1 6105 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) | |
| 2 | df-br 5144 | . 2 ⊢ (𝐵𝐴𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) | |
| 3 | 1, 2 | bitrdi 287 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 {csn 4626 〈cop 4632 class class class wbr 5143 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: elimasn 6108 inimasn 6176 dffv3 6902 fvimacnv 7073 fvrnressn 7181 elecg 8789 imasnopn 23698 imasncld 23699 imasncls 23700 ustelimasn 24231 blval2 24575 elbl4 24576 scutval 27845 iunsnima2 32631 1stpreimas 32715 opelco3 35775 funpartfv 35946 eltail 36375 elecALTV 38267 brtrclfv2 43740 frege77d 43759 dfafv23 47265 |
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