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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 43388 | . 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) |
3 | frege114.z | . . 3 ⊢ 𝑍 ∈ 𝑉 | |
4 | 3 | frege113 43397 | . 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 1534 ∈ wcel 2099 ∪ cun 3943 class class class wbr 5143 I cid 5570 ‘cfv 6543 t+ctcl 14959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-frege1 43211 ax-frege2 43212 ax-frege8 43230 ax-frege52a 43278 ax-frege52c 43309 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-opab 5206 df-id 5571 df-xp 5679 df-rel 5680 |
This theorem is referenced by: frege126 43410 |
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