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| Mirrors > Home > MPE Home > Th. List > elimasn1 | Structured version Visualization version GIF version | ||
| Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5144 and shorten proof. (Revised by BJ, 16-Oct-2024.) |
| Ref | Expression |
|---|---|
| elimasn1.1 | ⊢ 𝐵 ∈ V |
| elimasn1.2 | ⊢ 𝐶 ∈ V |
| Ref | Expression |
|---|---|
| elimasn1 | ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elimasn1.1 | . 2 ⊢ 𝐵 ∈ V | |
| 2 | elimasn1.2 | . 2 ⊢ 𝐶 ∈ V | |
| 3 | elimasng1 6105 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 {csn 4626 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: (None) |
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