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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege114 | Structured version Visualization version GIF version |
Description: If 𝑋 belongs to the 𝑅-sequence beginning with 𝑍, then 𝑍 belongs to the 𝑅-sequence beginning with 𝑋 or 𝑋 follows 𝑍 in the 𝑅-sequence. Proposition 114 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege114.x | ⊢ 𝑋 ∈ 𝑈 |
frege114.z | ⊢ 𝑍 ∈ 𝑉 |
Ref | Expression |
---|---|
frege114 | ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege114.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
2 | 1 | frege104 43870 | . 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) |
3 | frege114.z | . . 3 ⊢ 𝑍 ∈ 𝑉 | |
4 | 3 | frege113 43879 | . 2 ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))) |
5 | 2, 4 | ax-mp 5 | 1 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2103 ∪ cun 3968 class class class wbr 5169 I cid 5596 ‘cfv 6572 t+ctcl 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-frege1 43693 ax-frege2 43694 ax-frege8 43712 ax-frege52a 43760 ax-frege52c 43791 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ifp 1064 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 |
This theorem is referenced by: frege126 43892 |
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