Theorem dffrege99 41188

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	⊢ 𝑍 ∈ 𝑈

Assertion

Ref	Expression
dffrege99	⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem dffrege99

Step	Hyp	Ref	Expression
1		brun 5090	. 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍))
2		df-or 848	. 2 ⊢ ((𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍))
3		frege99.z	. . . . . 6 ⊢ 𝑍 ∈ 𝑈
4	3	elexi 3417	. . . . 5 ⊢ 𝑍 ∈ V
5	4	ideq 5706	. . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍)
6		eqcom 2743	. . . 4 ⊢ (𝑋 = 𝑍 ↔ 𝑍 = 𝑋)
7	5, 6	bitri 278	. . 3 ⊢ (𝑋 I 𝑍 ↔ 𝑍 = 𝑋)
8	7	imbi2i 339	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))
9	1, 2, 8	3bitrri 301	1 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 847   = wceq 1543   ∈ wcel 2112   ∪ cun 3851   class class class wbr 5039   I cid 5439  ‘cfv 6358  t+ctcl 14513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543

This theorem is referenced by:  frege100  41189  frege105  41194