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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 42313 | . 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) |
3 | frege114.z | . . 3 ⊢ 𝑍 ∈ 𝑉 | |
4 | 3 | frege113 42322 | . 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 1542 ∈ wcel 2107 ∪ cun 3913 class class class wbr 5110 I cid 5535 ‘cfv 6501 t+ctcl 14877 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-frege1 42136 ax-frege2 42137 ax-frege8 42155 ax-frege52a 42203 ax-frege52c 42234 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 |
This theorem is referenced by: frege126 42335 |
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