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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege74 | Structured version Visualization version GIF version |
Description: If 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then every result of a application of the procedure 𝑅 to 𝑋 has the property 𝐴. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege74.x | ⊢ 𝑋 ∈ 𝑈 |
frege74.y | ⊢ 𝑌 ∈ 𝑉 |
Ref | Expression |
---|---|
frege74 | ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege74.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
2 | frege74.y | . . 3 ⊢ 𝑌 ∈ 𝑉 | |
3 | 1, 2 | frege72 39070 | . 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
4 | ax-frege8 38944 | . 2 ⊢ ((𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) → (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 class class class wbr 4874 hereditary whe 38907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 ax-frege1 38925 ax-frege2 38926 ax-frege8 38944 ax-frege52a 38992 ax-frege58b 39036 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-cnv 5351 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-he 38908 |
This theorem is referenced by: frege81 39079 |
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