Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5949 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ ◡𝐴)) | |
3 | df-br 5059 | . . . 4 ⊢ (𝐵◡𝐴𝐶 ↔ 〈𝐵, 𝐶〉 ∈ ◡𝐴) | |
4 | 2, 3 | syl6bbr 290 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶)) |
5 | brcnvg 5744 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵)) | |
6 | 4, 5 | bitrd 280 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
7 | 1, 6 | mpan2 687 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 Vcvv 3495 {csn 4559 〈cop 4565 class class class wbr 5058 ◡ccnv 5548 “ cima 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 |
This theorem is referenced by: epini 5953 iniseg 5954 dfco2a 6093 elpred 6155 isomin 7079 isoini 7080 fnse 7818 infxpenlem 9428 fpwwe2lem8 10048 fpwwe2lem12 10052 fpwwe2lem13 10053 fpwwe2 10054 canth4 10058 canthwelem 10061 pwfseqlem4 10073 fz1isolem 13809 itg1addlem4 24229 elnlfn 29633 pw2f1ocnv 39514 |
Copyright terms: Public domain | W3C validator |