Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dffrege99 Structured version   Visualization version   GIF version

Theorem dffrege99 42713
Description: If 𝑍 is identical with 𝑋 or follows 𝑋 in the 𝑅 -sequence, then we say : "𝑍 belongs to the 𝑅-sequence beginning with 𝑋 " or "𝑋 belongs to the 𝑅-sequence ending with 𝑍". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)
Hypothesis
Ref Expression
frege99.z 𝑍𝑈
Assertion
Ref Expression
dffrege99 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem dffrege99
StepHypRef Expression
1 brun 5200 . 2 (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
2 df-or 847 . 2 ((𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
3 frege99.z . . . . . 6 𝑍𝑈
43elexi 3494 . . . . 5 𝑍 ∈ V
54ideq 5853 . . . 4 (𝑋 I 𝑍𝑋 = 𝑍)
6 eqcom 2740 . . . 4 (𝑋 = 𝑍𝑍 = 𝑋)
75, 6bitri 275 . . 3 (𝑋 I 𝑍𝑍 = 𝑋)
87imbi2i 336 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
91, 2, 83bitrri 298 1 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 846   = wceq 1542  wcel 2107  cun 3947   class class class wbr 5149   I cid 5574  cfv 6544  t+ctcl 14932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684
This theorem is referenced by:  frege100  42714  frege105  42719
  Copyright terms: Public domain W3C validator