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Mirrors > Home > MPE Home > Th. List > elimasni | Structured version Visualization version GIF version |
Description: Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.) |
Ref | Expression |
---|---|
elimasni | ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4330 | . . . . 5 ⊢ ¬ 𝐶 ∈ ∅ | |
2 | snprc 4721 | . . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 215 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 3 | imaeq2d 6059 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅)) |
5 | ima0 6076 | . . . . . . 7 ⊢ (𝐴 “ ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2788 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅) |
7 | 6 | eleq2d 2819 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅)) |
8 | 1, 7 | mtbiri 326 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵})) |
9 | 8 | con4i 114 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V) |
10 | elex 3492 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V) | |
11 | 9, 10 | jca 512 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
12 | elimasng1 6085 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) | |
13 | 12 | biimpd 228 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)) |
14 | 11, 13 | mpcom 38 | 1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 {csn 4628 class class class wbr 5148 “ cima 5679 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: dffv2 6986 poimirlem2 36576 poimirlem23 36597 |
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