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| Mirrors > Home > MPE Home > Th. List > eliniseg | Structured version Visualization version GIF version | ||
| Description: Membership in the inverse image of a singleton. An application is to express initial segments for an order relation. See for example Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| eliniseg.1 | ⊢ 𝐶 ∈ V |
| Ref | Expression |
|---|---|
| eliniseg | ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eliniseg.1 | . 2 ⊢ 𝐶 ∈ V | |
| 2 | elinisegg 6111 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Vcvv 3480 {csn 4626 class class class wbr 5143 ◡ccnv 5684 “ 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-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-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: epin 6113 iniseg 6115 dfco2a 6266 isomin 7357 isoini 7358 fnse 8158 infxpenlem 10053 fpwwe2lem7 10677 fpwwe2lem11 10681 fpwwe2lem12 10682 fpwwe2 10683 canth4 10687 canthwelem 10690 pwfseqlem4 10702 fz1isolem 14500 itg1addlem4 25734 elnlfn 31947 pw2f1ocnv 43049 relpmin 44973 |
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