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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege106 | Structured version Visualization version GIF version |
Description: Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege103.z | ⊢ 𝑍 ∈ 𝑉 |
Ref | Expression |
---|---|
frege106 | ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege103.z | . . 3 ⊢ 𝑍 ∈ 𝑉 | |
2 | 1 | frege105 41557 | . 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) |
3 | frege37 41429 | . 2 ⊢ (((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍)) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2106 ∪ cun 3884 class class class wbr 5073 I cid 5483 ‘cfv 6426 t+ctcl 14706 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 ax-frege1 41379 ax-frege2 41380 ax-frege8 41398 ax-frege28 41419 ax-frege31 41423 ax-frege52a 41446 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5074 df-opab 5136 df-id 5484 df-xp 5590 df-rel 5591 |
This theorem is referenced by: frege107 41559 |
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