MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpredim Structured version   Visualization version   GIF version

Theorem elpredim 6153
Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
Hypothesis
Ref Expression
elpredim.1 𝑋 ∈ V
Assertion
Ref Expression
elpredim (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Proof of Theorem elpredim
StepHypRef Expression
1 df-pred 6141 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 4172 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elpredim.1 . . . . 5 𝑋 ∈ V
4 elimasng 5948 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
5 opelcnvg 5744 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ 𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
64, 5bitrd 281 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
73, 6mpan 688 . . . 4 (𝑌 ∈ (𝑅 “ {𝑋}) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
87ibi 269 . . 3 (𝑌 ∈ (𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9 df-br 5058 . . 3 (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
108, 9sylibr 236 . 2 (𝑌 ∈ (𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
112, 10simplbiim 507 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2108  Vcvv 3493  {csn 4559  cop 4565   class class class wbr 5057  ccnv 5547  cima 5551  Predcpred 6140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141
This theorem is referenced by:  predbrg  6161  preddowncl  6168  trpredrec  33070
  Copyright terms: Public domain W3C validator