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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 5108 | . 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍)) | |
2 | df-or 842 | . 2 ⊢ ((𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍)) | |
3 | frege99.z | . . . . . 6 ⊢ 𝑍 ∈ 𝑈 | |
4 | 3 | elexi 3511 | . . . . 5 ⊢ 𝑍 ∈ V |
5 | 4 | ideq 5716 | . . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍) |
6 | eqcom 2825 | . . . 4 ⊢ (𝑋 = 𝑍 ↔ 𝑍 = 𝑋) | |
7 | 5, 6 | bitri 276 | . . 3 ⊢ (𝑋 I 𝑍 ↔ 𝑍 = 𝑋) |
8 | 7 | imbi2i 337 | . 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋)) |
9 | 1, 2, 8 | 3bitrri 299 | 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 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 class class class wbr 5057 I cid 5452 ‘cfv 6348 t+ctcl 14333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 |
This theorem is referenced by: frege100 40187 frege105 40192 |
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