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Mirrors > Home > MPE Home > Th. List > elimasn | 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.) (Proof shortened by BJ, 16-Oct-2024.) TODO: replace existing usages by usages of elimasn1 6087, remove, and relabel elimasn1 6087 to "elimasn". |
Ref | Expression |
---|---|
elimasn.1 | ⊢ 𝐵 ∈ V |
elimasn.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elimasn | ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimasn.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | elimasn.2 | . 2 ⊢ 𝐶 ∈ V | |
3 | elimasng 6088 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 Vcvv 3475 {csn 4629 ⟨cop 4635 “ cima 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 |
This theorem is referenced by: elimasngOLD 6090 dfco2 6245 dfco2a 6246 ressn 6285 funfvima3 7238 frxp 8112 frxp2 8130 frxp3 8137 marypha1lem 9428 gsum2dlem1 19838 gsum2dlem2 19839 gsum2d 19840 gsum2d2 19842 ovoliunlem1 25019 dmscut 27312 scutf 27313 iunsnima 31847 dfcnv2 31901 gsummpt2co 32200 gsummpt2d 32201 funpartfun 34915 areaquad 41965 dffrege76 42690 frege97 42711 frege98 42712 frege109 42723 frege110 42724 frege131 42745 frege133 42747 |
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