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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 5109 | . 2 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍)) | |
2 | df-or 844 | . 2 ⊢ ((𝑋(t+‘𝑅)𝑍 ∨ 𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 I 𝑍)) | |
3 | frege99.z | . . . . . 6 ⊢ 𝑍 ∈ 𝑈 | |
4 | 3 | elexi 3513 | . . . . 5 ⊢ 𝑍 ∈ V |
5 | 4 | ideq 5717 | . . . 4 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍) |
6 | eqcom 2828 | . . . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 class class class wbr 5058 I cid 5453 ‘cfv 6349 t+ctcl 14339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 |
This theorem is referenced by: frege100 40302 frege105 40307 |
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