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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 5265. 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 43352, which is the virtual deduction proof sspwtr 43351 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 3971 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴)) | |
2 | 1 | adantld 491 | . . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝒫 𝐴)) |
3 | elpwi 4603 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴) | |
4 | 2, 3 | syl6 35 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴)) |
5 | simpl 483 | . . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)) |
7 | ssel 3971 | . . . 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 1931 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) |
10 | dftr2 5260 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
11 | 9, 10 | sylibr 233 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 ∈ wcel 2106 ⊆ wss 3944 𝒫 cpw 4596 Tr wtr 5258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-pw 4598 df-uni 4902 df-tr 5259 |
This theorem is referenced by: (None) |
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