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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege111d | Structured version Visualization version GIF version |
Description: If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 40311. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege111d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege111d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
frege111d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege111d.c | ⊢ (𝜑 → 𝐶 ∈ V) |
frege111d.ac | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) |
frege111d.cb | ⊢ (𝜑 → 𝐶𝑅𝐵) |
Ref | Expression |
---|---|
frege111d | ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege111d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | frege111d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
3 | frege111d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
4 | frege111d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) | |
5 | frege111d.ac | . . 3 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) | |
6 | frege111d.cb | . . 3 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | frege108d 40092 | . 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) |
8 | 7 | frege114d 40094 | 1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∨ w3o 1081 = wceq 1531 ∈ wcel 2108 Vcvv 3493 class class class wbr 5057 ‘cfv 6348 t+ctcl 14337 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-trcl 14339 |
This theorem is referenced by: (None) |
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