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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 40159 | . 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
4 | ax-frege8 40033 | . 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 2105 class class class wbr 5057 hereditary whe 39996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-frege1 40014 ax-frege2 40015 ax-frege8 40033 ax-frege52a 40081 ax-frege58b 40125 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-he 39997 |
This theorem is referenced by: frege81 40168 |
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