Theorem dffrege99 44486

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 5145	. 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍))
2		df-or 857	. 2 ⊢ ((𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍))
3		frege99.z	. . . . . 6 ⊢ 𝑍 ∈ 𝑈
4	3	elexi 3470	. . . . 5 ⊢ 𝑍 ∈ V
5	4	ideq 5817	. . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍)
6		eqcom 2763	. . . 4 ⊢ (𝑋 = 𝑍 ↔ 𝑍 = 𝑋)
7	5, 6	bitri 277	. . 3 ⊢ (𝑋 I 𝑍 ↔ 𝑍 = 𝑋)
8	7	imbi2i 338	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))
9	1, 2, 8	3bitrri 300	1 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 856   = wceq 1554   ∈ wcel 2136   ∪ cun 3897   class class class wbr 5094   I cid 5534  ‘cfv 6510  t+ctcl 14988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647

This theorem is referenced by:  frege100  44487  frege105  44492