|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 4338 | . . . . 5 ⊢ ¬ 𝐶 ∈ ∅ | |
| 2 | snprc 4717 | . . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
| 3 | 2 | biimpi 216 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) | 
| 4 | 3 | imaeq2d 6078 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅)) | 
| 5 | ima0 6095 | . . . . . . 7 ⊢ (𝐴 “ ∅) = ∅ | |
| 6 | 4, 5 | eqtrdi 2793 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅) | 
| 7 | 6 | eleq2d 2827 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅)) | 
| 8 | 1, 7 | mtbiri 327 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵})) | 
| 9 | 8 | con4i 114 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V) | 
| 10 | elex 3501 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V) | |
| 11 | 9, 10 | jca 511 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | 
| 12 | elimasng1 6105 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) | |
| 13 | 12 | biimpd 229 | . 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 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: dffv2 7004 poimirlem2 37629 poimirlem23 37650 | 
| Copyright terms: Public domain | W3C validator |