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Mirrors > Home > MPE Home > Th. List > eliniseg | Structured version Visualization version GIF version |
Description: Membership in an initial segment. The idiom (◡𝐴 “ {𝐵}), meaning {𝑥 ∣ 𝑥𝐴𝐵}, is used to specify an initial segment in (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 | elimasng 5410 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ ◡𝐴)) | |
3 | df-br 4584 | . . . 4 ⊢ (𝐵◡𝐴𝐶 ↔ 〈𝐵, 𝐶〉 ∈ ◡𝐴) | |
4 | 2, 3 | syl6bbr 277 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶)) |
5 | brcnvg 5225 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵)) | |
6 | 4, 5 | bitrd 267 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
7 | 1, 6 | mpan2 703 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 “ 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: epini 5414 iniseg 5415 dfco2a 5552 elpred 5610 isomin 6487 isoini 6488 fnse 7181 infxpenlem 8719 fpwwe2lem8 9338 fpwwe2lem12 9342 fpwwe2lem13 9343 fpwwe2 9344 canth4 9348 canthwelem 9351 pwfseqlem4 9363 fz1isolem 13102 itg1addlem4 23272 elnlfn 28171 pw2f1ocnv 36622 |
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