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Theorem eliniseg 5482
Description: Membership in an initial segment. The idiom (𝐴 “ {𝐵}), meaning {𝑥𝑥𝐴𝐵}, 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 (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Proof of Theorem eliniseg
StepHypRef Expression
1 eliniseg.1 . 2 𝐶 ∈ V
2 elimasng 5479 . . . 4 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
3 df-br 4645 . . . 4 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
42, 3syl6bbr 278 . . 3 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
5 brcnvg 5292 . . 3 ((𝐵𝑉𝐶 ∈ V) → (𝐵𝐴𝐶𝐶𝐴𝐵))
64, 5bitrd 268 . 2 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
71, 6mpan2 706 1 (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1988  Vcvv 3195  {csn 4168  cop 4174   class class class wbr 4644  ccnv 5103  cima 5107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117
This theorem is referenced by:  epini  5483  iniseg  5484  dfco2a  5623  elpred  5681  isomin  6572  isoini  6573  fnse  7279  infxpenlem  8821  fpwwe2lem8  9444  fpwwe2lem12  9448  fpwwe2lem13  9449  fpwwe2  9450  canth4  9454  canthwelem  9457  pwfseqlem4  9469  fz1isolem  13228  itg1addlem4  23447  elnlfn  28757  pw2f1ocnv  37423
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