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Mirrors > Home > MPE Home > Th. List > Mathboxes > trsspwALT3 | Structured version Visualization version GIF version |
Description: Short predicate calculus proof of the left-to-right implication of dftr4 5267. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 44323, which is the virtual deduction proof trsspwALT 44322 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
trsspwALT3 | ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trss 5271 | . . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
2 | vex 3467 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 4602 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
4 | 1, 3 | imbitrrdi 251 | . 2 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) |
5 | 4 | ssrdv 3978 | 1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3939 𝒫 cpw 4598 Tr wtr 5260 |
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-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-v 3465 df-ss 3956 df-pw 4600 df-uni 4904 df-tr 5261 |
This theorem is referenced by: (None) |
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