![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dffrege99 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frege99.z | ⊢ 𝑍 ∈ 𝑈 |
Ref | Expression |
---|---|
dffrege99 | ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brun 5161 | . 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍)) | |
2 | df-or 847 | . 2 ⊢ ((𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍)) | |
3 | frege99.z | . . . . . 6 ⊢ 𝑍 ∈ 𝑈 | |
4 | 3 | elexi 3467 | . . . . 5 ⊢ 𝑍 ∈ V |
5 | 4 | ideq 5813 | . . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍) |
6 | eqcom 2744 | . . . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∪ cun 3913 class class class wbr 5110 I cid 5535 ‘cfv 6501 t+ctcl 14877 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 |
This theorem is referenced by: frege100 42309 frege105 42314 |
Copyright terms: Public domain | W3C validator |