eliniseg - Metamath Proof Explorer

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Theorem eliniseg 5494

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 5491	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ ^◡𝐴))
3		df-br 4654	. . . 4 ⊢ (𝐵^◡𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ^◡𝐴)
4	2, 3	syl6bbr 278	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐵^◡𝐴𝐶))
5		brcnvg 5303	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐵^◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
6	4, 5	bitrd 268	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
7	1, 6	mpan2 707	1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (^◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 {csn 4177 ⟨cop 4183 class class class wbr 4653 ^◡ccnv 5113 “ cima 5117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127

This theorem is referenced by: epini 5495 iniseg 5496 dfco2a 5635 elpred 5693 isomin 6587 isoini 6588 fnse 7294 infxpenlem 8836 fpwwe2lem8 9459 fpwwe2lem12 9463 fpwwe2lem13 9464 fpwwe2 9465 canth4 9469 canthwelem 9472 pwfseqlem4 9484 fz1isolem 13245 itg1addlem4 23466 elnlfn 28787 pw2f1ocnv 37604

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