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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 5044 and shorten. (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 5943 | . 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 209 ∈ wcel 2110 Vcvv 3401 {csn 4531 class class class wbr 5043 “ cima 5543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 |
This theorem is referenced by: (None) |
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