Theorem dffrege99 38256

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 4703	. 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍))
2		df-or 385	. 2 ⊢ ((𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍))
3		frege99.z	. . . . . 6 ⊢ 𝑍 ∈ 𝑈
4	3	elexi 3213	. . . . 5 ⊢ 𝑍 ∈ V
5	4	ideq 5274	. . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍)
6		eqcom 2629	. . . 4 ⊢ (𝑋 = 𝑍 ↔ 𝑍 = 𝑋)
7	5, 6	bitri 264	. . 3 ⊢ (𝑋 I 𝑍 ↔ 𝑍 = 𝑋)
8	7	imbi2i 326	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))
9	1, 2, 8	3bitrri 287	1 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   class class class wbr 4653   I cid 5023  ‘cfv 5888  t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121

This theorem is referenced by:  frege100  38257  frege105  38262