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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 4344 | . . . . 5 ⊢ ¬ 𝐶 ∈ ∅ | |
2 | snprc 4722 | . . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 216 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 3 | imaeq2d 6080 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅)) |
5 | ima0 6097 | . . . . . . 7 ⊢ (𝐴 “ ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2791 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅) |
7 | 6 | eleq2d 2825 | . . . . 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 3499 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V) | |
11 | 9, 10 | jca 511 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
12 | elimasng1 6107 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: dffv2 7004 poimirlem2 37609 poimirlem23 37630 |
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