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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 3878 | . . . . 5 ⊢ ¬ 𝐶 ∈ ∅ | |
2 | snprc 4197 | . . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | 2 | biimpi 205 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅) |
4 | 3 | imaeq2d 5385 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅)) |
5 | ima0 5400 | . . . . . . 7 ⊢ (𝐴 “ ∅) = ∅ | |
6 | 4, 5 | syl6eq 2660 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅) |
7 | 6 | eleq2d 2673 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅)) |
8 | 1, 7 | mtbiri 316 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵})) |
9 | 8 | con4i 112 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V) |
10 | elex 3185 | . . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V) | |
11 | 9, 10 | jca 553 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
12 | elimasng 5410 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴)) | |
13 | df-br 4584 | . . . 4 ⊢ (𝐵𝐴𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) | |
14 | 12, 13 | syl6bbr 277 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)) |
15 | 14 | biimpd 218 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)) |
16 | 11, 15 | mpcom 37 | 1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 〈cop 4131 class class class wbr 4583 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: dffv2 6181 poimirlem2 32581 poimirlem23 32602 |
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