Theorem dffrege99 43974

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 5140	. 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 3457	. . . . 5 ⊢ 𝑍 ∈ V
5	4	ideq 5790	. . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍)
6		eqcom 2737	. . . 4 ⊢ (𝑋 = 𝑍 ↔ 𝑍 = 𝑋)
7	5, 6	bitri 275	. . 3 ⊢ (𝑋 I 𝑍 ↔ 𝑍 = 𝑋)
8	7	imbi2i 336	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))
9	1, 2, 8	3bitrri 298	1 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2110   ∪ cun 3898   class class class wbr 5089   I cid 5508  ‘cfv 6477  t+ctcl 14884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621

This theorem is referenced by:  frege100  43975  frege105  43980