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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege71 | Structured version Visualization version GIF version |
Description: Lemma for frege72 43368. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege71.x | ⊢ 𝑋 ∈ 𝑉 |
Ref | Expression |
---|---|
frege71 | ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege71.x | . . 3 ⊢ 𝑋 ∈ 𝑉 | |
2 | 1 | frege70 43366 | . 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))) |
3 | frege19 43257 | . 2 ⊢ ((𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))) → ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∈ wcel 2098 class class class wbr 5150 hereditary whe 43205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-frege1 43223 ax-frege2 43224 ax-frege8 43242 ax-frege52a 43290 ax-frege58b 43334 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5151 df-opab 5213 df-xp 5686 df-cnv 5688 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-he 43206 |
This theorem is referenced by: frege72 43368 |
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