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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwtrALT2 | Structured version Visualization version GIF version |
Description: Short predicate calculus proof of the right-to-left implication of dftr4 5177. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 41176, which is the virtual deduction proof sspwtr 41175 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspwtrALT2 | ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3961 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴)) | |
2 | 1 | adantld 493 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝒫 𝐴)) |
3 | elpwi 4548 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴) | |
4 | 2, 3 | syl6 35 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴)) |
5 | simpl 485 | . . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)) |
7 | ssel 3961 | . . . 4 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴)) | |
8 | 4, 6, 7 | syl6c 70 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) |
9 | 8 | alrimivv 1929 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) |
10 | dftr2 5174 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
11 | 9, 10 | sylibr 236 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 Tr wtr 5172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4839 df-tr 5173 |
This theorem is referenced by: (None) |
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