eliniseg - Metamath Proof Explorer

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Theorem eliniseg 5711

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.)

**Hypothesis**
Ref	Expression
eliniseg.1	⊢ 𝐶 ∈ V

**Assertion**
Ref	Expression
eliniseg	⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

**Proof of Theorem eliniseg**
Step	Hyp	Ref	Expression
1		eliniseg.1	. 2 ⊢ 𝐶 ∈ V
2		elimasng 5708	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ ^◡𝐴))
3		df-br 4844	. . . 4 ⊢ (𝐵^◡𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ^◡𝐴)
4	2, 3	syl6bbr 281	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐵^◡𝐴𝐶))
5		brcnvg 5505	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐵^◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
6	4, 5	bitrd 271	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
7	1, 6	mpan2 683	1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∈ wcel 2157 Vcvv 3385 {csn 4368 ⟨cop 4374 class class class wbr 4843 ^◡ccnv 5311 “ cima 5315

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325

This theorem is referenced by: epini 5712 iniseg 5713 dfco2a 5854 elpred 5911 isomin 6815 isoini 6816 fnse 7531 infxpenlem 9122 fpwwe2lem8 9747 fpwwe2lem12 9751 fpwwe2lem13 9752 fpwwe2 9753 canth4 9757 canthwelem 9760 pwfseqlem4 9772 fz1isolem 13494 itg1addlem4 23807 elnlfn 29312 pw2f1ocnv 38389

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