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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 37282 | . 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) |
3 | frege37 37154 | . 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 1475 ∈ wcel 1977 ∪ cun 3538 class class class wbr 4583 I cid 4948 ‘cfv 5804 t+ctcl 13572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-frege1 37104 ax-frege2 37105 ax-frege8 37123 ax-frege28 37144 ax-frege31 37148 ax-frege52a 37171 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 |
This theorem is referenced by: frege107 37284 |
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